Papers
Preprints
- On the chromatic number of the union of comparability graphs (with Maria Chudnovsky, Linda Cook, James Davies, and Seokbeom Kim김석범), June 2026.Resolving in a strong sense an old problem of Gyárfás from the 1980s on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs.
On the chromatic number of the union of comparability graphs
Resolving in a strong sense an old problem of Gyárfás from the 1980s on the union of two perfect graphs, we prove that for every pair of positive integers $d$ and $k$, there is a graph $G$ with clique number $k$ and chromatic number $k^d$ that is the union of $d$ comparability graphs.
- Branch-width of represented matroids in matrix multiplication time (with Mujin Choi최무진 and Tuukka Korhonen), May 2026.For an $n$-element matroid $M$ given by an $n \times n$ matrix representation over a finite field $\mathbb F$ and an integer $k$, we present an $(O_{k,\mathbb F}(n^2)+O(n^ω))$-time algorithm that either finds a branch-decomposition of $M$ of width at most $k$, or confirms that the branch-width of $M$ is more than $k$, where $ω< 2.3714$ is the matrix multiplication exponent, and the $O_{k,\mathbb F}(\cdot)$-notation hides factors that depend on $k$ and $\mathbb F$ in a computable manner. All previous algorithms including Hliněný and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, and Oum [SIAM J. Discrete Math. (2021)] run in at least $Ω(n^3)$ time. Moreover, if the input matrix representation is given by a standard form, our algorithm runs in $O_{k,\mathbb F}(n^2)$-time, since $O(n^ω)$-time is only needed for finding a standard form of the input matrix. When $M$ is given by an $m \times n$ matrix, the overhead for finding a standard form is $O(mn \min(m,n)^{ω-2})$. As corollaries, we obtain faster algorithms for rank-width of directed graphs and path-width of matroids represented over a fixed finite field. Furthermore, we also present an approximation algorithm for finding branch-width that works on infinite fields provided that the input matrix is of a standard form and contains a bounded number of distinct values of entries. To suggest that our algorithm is optimal, we observe that for every field $\mathbb F$, deciding whether the branch-width of a matroid represented over $\mathbb F$ is $0$ is as hard as deciding whether a square matrix over $\mathbb F$ is singular. Under the assumption that singularity testing requires $Ω(n^ω)$-time, this implies that the overhead of $O(n^ω)$ is unavoidable. We also show strengthenings of this observation to rule out some approximations under this assumption.
Branch-width of represented matroids in matrix multiplication time
For an $n$-element matroid $M$ given by an $n \times n$ matrix representation over a finite field $\mathbb F$ and an integer $k$, we present an $(O_{k,\mathbb F}(n^2)+O(n^ω))$-time algorithm that either finds a branch-decomposition of $M$ of width at most $k$, or confirms that the branch-width of $M$ is more than $k$, where $ω< 2.3714$ is the matrix multiplication exponent, and the $O_{k,\mathbb F}(\cdot)$-notation hides factors that depend on $k$ and $\mathbb F$ in a computable manner. All previous algorithms including Hliněný and Oum [SIAM J. Comput. (2008)] and Jeong, Kim, and Oum [SIAM J. Discrete Math. (2021)] run in at least $Ω(n^3)$ time. Moreover, if the input matrix representation is given by a standard form, our algorithm runs in $O_{k,\mathbb F}(n^2)$-time, since $O(n^ω)$-time is only needed for finding a standard form of the input matrix. When $M$ is given by an $m \times n$ matrix, the overhead for finding a standard form is $O(mn \min(m,n)^{ω-2})$. As corollaries, we obtain faster algorithms for rank-width of directed graphs and path-width of matroids represented over a fixed finite field. Furthermore, we also present an approximation algorithm for finding branch-width that works on infinite fields provided that the input matrix is of a standard form and contains a bounded number of distinct values of entries. To suggest that our algorithm is optimal, we observe that for every field $\mathbb F$, deciding whether the branch-width of a matroid represented over $\mathbb F$ is $0$ is as hard as deciding whether a square matrix over $\mathbb F$ is singular. Under the assumption that singularity testing requires $Ω(n^ω)$-time, this implies that the overhead of $O(n^ω)$ is unavoidable. We also show strengthenings of this observation to rule out some approximations under this assumption.
- Ramsey-type χ-bounds for χ-bounded graph classes (with Tung H. Nguyen), May 2026.We prove that for every path $P$, the class of graphs with no induced $P$ and no induced four-cycle $C_4$ is linearly $χ$-bounded. More generally, we ask for which pairs $\{T,H\}$ where $T$ is a forest and $H$ is a complete multipartite graph, every graph $G$ with no induced $T$ and no induced $H$ has chromatic number at most $C \cdot R(α(H),ω(G)+1)$ for some constant $C$ depending only on $T$ and $H$, where $R(\cdot,\cdot)$ denotes the usual Ramsey numbers. We show that this holds in the following two instances, which strengthen the case $T=P$ and $H=C_4$ mentioned above: (1) every component of $T$ is a broom and $H$ is complete multipartite; or (2) $T$ is a forest and $H$ is complete bipartite. These two unify and substantially extend a number of previous results on linear and polynomial $χ$-boundedness for various graph classes. For case (2), we also provide a new proof (with better bounds) of a recent result of Fox, Nenadov, and Pham on the existence of an induced copy of a fixed tree in a graph satisfying certain sparsity conditions.
Ramsey-type χ-bounds for χ-bounded graph classes
We prove that for every path $P$, the class of graphs with no induced $P$ and no induced four-cycle $C_4$ is linearly $χ$-bounded. More generally, we ask for which pairs $\{T,H\}$ where $T$ is a forest and $H$ is a complete multipartite graph, every graph $G$ with no induced $T$ and no induced $H$ has chromatic number at most $C \cdot R(α(H),ω(G)+1)$ for some constant $C$ depending only on $T$ and $H$, where $R(\cdot,\cdot)$ denotes the usual Ramsey numbers. We show that this holds in the following two instances, which strengthen the case $T=P$ and $H=C_4$ mentioned above: (1) every component of $T$ is a broom and $H$ is complete multipartite; or (2) $T$ is a forest and $H$ is complete bipartite. These two unify and substantially extend a number of previous results on linear and polynomial $χ$-boundedness for various graph classes. For case (2), we also provide a new proof (with better bounds) of a recent result of Fox, Nenadov, and Pham on the existence of an induced copy of a fixed tree in a graph satisfying certain sparsity conditions.
- Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions (with Marek Sokołowski), March 2026.We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining $0$ on the empty set. For a connectivity function $f$ on an $n$-element set $V$ and an integer $k\ge 0$, we show that the family of all sets $X\subseteq V$ with $f(X)=k$ admits a polynomial-size representation: it can be described by a list of at most $O(n^{4k})$ items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets $X$ with $f(X)=k$. We also give an algorithm that constructs this representation in time $O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})$, where $γ$ is the oracle time to evaluate $f$. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, arXiv, 2025] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed $k$, we obtain a polynomial-time algorithm for finding a set $A$ with $f(A)=k$ and a prescribed cardinality constraint on $A$.
Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions
We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining $0$ on the empty set. For a connectivity function $f$ on an $n$-element set $V$ and an integer $k\ge 0$, we show that the family of all sets $X\subseteq V$ with $f(X)=k$ admits a polynomial-size representation: it can be described by a list of at most $O(n^{4k})$ items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets $X$ with $f(X)=k$. We also give an algorithm that constructs this representation in time $O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})$, where $γ$ is the oracle time to evaluate $f$. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, arXiv, 2025] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed $k$, we obtain a polynomial-time algorithm for finding a set $A$ with $f(A)=k$ and a prescribed cardinality constraint on $A$.
- Branch-width of connectivity functions is fixed-parameter tractable (with Tuukka Korhonen), January 2026.A connectivity function on a finite set $V$ is a symmetric submodular function $f \colon 2^V \to \mathbb{Z}$ with $f(\emptyset)=0$. We prove that finding a branch-decomposition of width at most $k$ for a connectivity function given by an oracle is fixed-parameter tractable (FPT), by providing an algorithm of running time $2^{O(k^2)} γn^6 \log n$, where $γ$ is the time to compute $f(X)$ for any set $X$, and $n = |V|$. This improves the previous algorithm by Oum and Seymour [J. Combin. Theory Ser. B, 2007], which runs in time $γn^{O(k)}$. Our algorithm can be applied to rank-width of graphs, branch-width of matroids, branch-width of (hyper)graphs, and carving-width of graphs. This resolves an open problem asked by Hliněný [SIAM J. Comput., 2005], who asked whether branch-width of matroids given by the rank oracle is fixed-parameter tractable. Furthermore, our algorithm improves the best known dependency on $k$ in the running times of FPT algorithms for graph branch-width, rank-width, and carving-width.
Branch-width of connectivity functions is fixed-parameter tractable
A connectivity function on a finite set $V$ is a symmetric submodular function $f \colon 2^V \to \mathbb{Z}$ with $f(\emptyset)=0$. We prove that finding a branch-decomposition of width at most $k$ for a connectivity function given by an oracle is fixed-parameter tractable (FPT), by providing an algorithm of running time $2^{O(k^2)} γn^6 \log n$, where $γ$ is the time to compute $f(X)$ for any set $X$, and $n = |V|$. This improves the previous algorithm by Oum and Seymour [J. Combin. Theory Ser. B, 2007], which runs in time $γn^{O(k)}$. Our algorithm can be applied to rank-width of graphs, branch-width of matroids, branch-width of (hyper)graphs, and carving-width of graphs. This resolves an open problem asked by Hliněný [SIAM J. Comput., 2005], who asked whether branch-width of matroids given by the rank oracle is fixed-parameter tractable. Furthermore, our algorithm improves the best known dependency on $k$ in the running times of FPT algorithms for graph branch-width, rank-width, and carving-width.
- Blind cop-width and balanced minors of graphs (with Hector Buffière, Rutger Campbell, and Kevin Hendrey), November 2025.We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices in the graph. The blind cop-width is the minimum number of cops needed to catch the robber on a given graph. We link it with other known graph parameters defined in terms of pursuit-evasion games, and show a new lower bound with respect to treewidth. The proof introduces the notion of balanced minors, where all branch sets of a minor model have equal size.
Blind cop-width and balanced minors of graphs
We investigate a pursuit-evasion game on an undirected graph in which a robber, moving at a fixed constant speed, attempts to evade a team of cops who are blind to the robber's location and can quickly travel between any pair of vertices in the graph. The blind cop-width is the minimum number of cops needed to catch the robber on a given graph. We link it with other known graph parameters defined in terms of pursuit-evasion games, and show a new lower bound with respect to treewidth. The proof introduces the notion of balanced minors, where all branch sets of a minor model have equal size.
- The Erdős-Pósa property for circle graphs as vertex-minors (with Rutger Campbell, J. Pascal Gollin, Meike Hatzel, O-joung Kwon권오정, Rose McCarty, and Sebastian Wiederrecht), June 2025.We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$.
The Erdős-Pósa property for circle graphs as vertex-minors
We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$.
- Colouring t-perfect graphs (with Maria Chudnovsky, Linda Cook, James Davies, and Jane Tan), December 2024.Perfect graphs can be described as the graphs whose stable set polytopes are defined by their non-negativity and clique inequalities (including edge inequalities). In 1975, Chvátal defined an analogous class of t-perfect graphs, which are the graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. We show that t-perfect graphs are $199053$-colourable. This is the first finite bound on the chromatic number of t-perfect graphs and answers a question of Shepherd from 1995. Our proof also shows that every h-perfect graph with clique number $ω$ is $(ω+ 199050)$-colourable.
Colouring t-perfect graphs
Perfect graphs can be described as the graphs whose stable set polytopes are defined by their non-negativity and clique inequalities (including edge inequalities). In 1975, Chvátal defined an analogous class of t-perfect graphs, which are the graphs whose stable set polytopes are defined by their non-negativity, edge inequalities, and odd circuit inequalities. We show that t-perfect graphs are $199053$-colourable. This is the first finite bound on the chromatic number of t-perfect graphs and answers a question of Shepherd from 1995. Our proof also shows that every h-perfect graph with clique number $ω$ is $(ω+ 199050)$-colourable.
Journal Papers
Accepted
- Unavoidable pivot-minors in graphs of large rank-depth (with Jungho Ahn안정호, Kevin Hendrey, and O-joung Kwon권오정), SIAM J. Discrete Math., accepted, 2026.Shrub-depth and rank-depth are related graph parameters that are dense analogs of tree-depth. We prove that for every positive integer $t$, every graph of sufficiently large rank-depth contains a pivot-minor isomorphic to a path on $t$ vertices or a graph consisting of two disjoint cliques of size $t$ joined by a half graph. This answers an open problem raised by Kwon, McCarty, Oum, and Wollan in 2021.
Unavoidable pivot-minors in graphs of large rank-depth
Shrub-depth and rank-depth are related graph parameters that are dense analogs of tree-depth. We prove that for every positive integer $t$, every graph of sufficiently large rank-depth contains a pivot-minor isomorphic to a path on $t$ vertices or a graph consisting of two disjoint cliques of size $t$ joined by a half graph. This answers an open problem raised by Kwon, McCarty, Oum, and Wollan in 2021.
- Sharing tea on a graph (with J. Pascal Gollin, Kevin Hendrey, Hao Huang, Tony Huynh, Bojan Mohar, Wei-Hsuan Yu, Ningyuan Yang, and Xuding Zhu), Comb. Theory, accepted, June 2026.Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert. We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \in \mathbb{R}_{\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\mathbb{R}_{\geq 0}^{V(G)}$.
Sharing tea on a graph
Motivated by the analysis of consensus formation in the Deffuant model for social interaction, we consider the following procedure on a graph $G$. Initially, there is one unit of tea at a fixed vertex $r \in V(G)$, and all other vertices have no tea. At any time in the procedure, we can choose a connected subset of vertices $T$ and equalize the amount of tea among vertices in $T$. We prove that if $x \in V(G)$ is at distance $d$ from $r$, then $x$ will have at most $\frac{1}{d+1}$ units of tea during any step of the procedure. This bound is best possible and answers a question of Gantert. We also consider arbitrary initial weight distributions. For every finite graph $G$ and $w \in \mathbb{R}_{\geq 0}^{V(G)}$, we prove that the set of weight distributions reachable from $w$ is a compact subset of $\mathbb{R}_{\geq 0}^{V(G)}$.
2026
- Computing pivot-minors (with Konrad K. Dabrowski, François Dross, Jisu Jeong정지수, Mamadou Moustapha Kanté, O-joung Kwon권오정, and Daniël Paulusma), Algorithmica, 88:47, May 2026.A graph $G$ contains a graph $H$ as a pivot-minor if $H$ can be obtained from $G$ by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. Pivot-minors have mainly been studied from a structural perspective. In this paper we perform the first systematic computational complexity study of pivot-minors. We first prove that the Pivot-Minor problem, which asks if a given graph $G$ contains a pivot-minor isomorphic to a given graph $H$, is NP-complete. If $H$ is not part of the input, we denote the problem by $H$-Pivot-Minor. We give a certifying polynomial-time algorithm for $H$-Pivot-Minor when (1) $H$ is an induced subgraph of $P_3+tP_1$ for some integer $t\geq 0$, (2) $H=K_{1,t}$ for some integer $t\geq 1$, or (3) $|V(H)|\leq 4$ except when $H \in \{K_4,C_3+ P_1\}$. Let ${\cal F}_H$ be the set of induced-subgraph-minimal graphs that contain a pivot-minor isomorphic to $H$. To prove the above statement, we either show that there is an integer $c_H$ such that all graphs in ${\cal F}_H$ have at most $c_H$ vertices, or we determine ${\cal F}_H$ precisely, for each of the above cases.
Computing pivot-minors
A graph $G$ contains a graph $H$ as a pivot-minor if $H$ can be obtained from $G$ by applying a sequence of vertex deletions and edge pivots. Pivot-minors play an important role in the study of rank-width. Pivot-minors have mainly been studied from a structural perspective. In this paper we perform the first systematic computational complexity study of pivot-minors. We first prove that the Pivot-Minor problem, which asks if a given graph $G$ contains a pivot-minor isomorphic to a given graph $H$, is NP-complete. If $H$ is not part of the input, we denote the problem by $H$-Pivot-Minor. We give a certifying polynomial-time algorithm for $H$-Pivot-Minor when (1) $H$ is an induced subgraph of $P_3+tP_1$ for some integer $t\geq 0$, (2) $H=K_{1,t}$ for some integer $t\geq 1$, or (3) $|V(H)|\leq 4$ except when $H \in \{K_4,C_3+ P_1\}$. Let ${\cal F}_H$ be the set of induced-subgraph-minimal graphs that contain a pivot-minor isomorphic to $H$. To prove the above statement, we either show that there is an integer $c_H$ such that all graphs in ${\cal F}_H$ have at most $c_H$ vertices, or we determine ${\cal F}_H$ precisely, for each of the above cases.
- Fragile minor-monotone parameters under random edge perturbation (with Dong Yeap Kang강동엽, Mihyun Kang강미현, and Jaehoon Kim김재훈), European J. Combin., 133:104305, March 2026.We conduct a quantitative analysis of how many random edges need to be added to a base graph $H$ in order to significantly increase natural minor-monotone graph parameters of the resulting graph $R$. Specifically, we show that if $R$ is obtained from a connected graph $H$ by adding only a few random edges, the tree-width, genus, and Hadwiger number of $R$ become very large, irrespective of the structure of $H$.
Fragile minor-monotone parameters under random edge perturbation
We conduct a quantitative analysis of how many random edges need to be added to a base graph $H$ in order to significantly increase natural minor-monotone graph parameters of the resulting graph $R$. Specifically, we show that if $R$ is obtained from a connected graph $H$ by adding only a few random edges, the tree-width, genus, and Hadwiger number of $R$ become very large, irrespective of the structure of $H$.
- Reuniting χ-boundedness with polynomial χ-boundedness (with Maria Chudnovsky, Linda Cook, and James Davies), J. Combin. Theory Ser. B, 176:30-73, January 2026.A class $\mathcal{F}$ of graphs is $χ$-bounded if there is a function $f$ such that $χ(H)\le f(ω(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$ is polynomially $χ$-bounded. Esperet proposed a conjecture that every $χ$-bounded class of graphs is polynomially $χ$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $χ$-bounded but not polynomially $χ$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $\mathcal{C}$ of graphs is Pollyanna if $\mathcal{C}\cap \mathcal{F}$ is polynomially $χ$-bounded for every $χ$-bounded class $\mathcal{F}$ of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.
Reuniting χ-boundedness with polynomial χ-boundedness
A class $\mathcal{F}$ of graphs is $χ$-bounded if there is a function $f$ such that $χ(H)\le f(ω(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$ is polynomially $χ$-bounded. Esperet proposed a conjecture that every $χ$-bounded class of graphs is polynomially $χ$-bounded. This conjecture has been disproved; it has been shown that there are classes of graphs that are $χ$-bounded but not polynomially $χ$-bounded. Nevertheless, inspired by Esperet's conjecture, we introduce Pollyanna classes of graphs. A class $\mathcal{C}$ of graphs is Pollyanna if $\mathcal{C}\cap \mathcal{F}$ is polynomially $χ$-bounded for every $χ$-bounded class $\mathcal{F}$ of graphs. We prove that several classes of graphs are Pollyanna and also present some proper classes of graphs that are not Pollyanna.
2025
- Linear bounds on treewidth in terms of excluded planar minors (with J. Pascal Gollin, Kevin Hendrey, and Bruce Reed), Electron. J. Combin., 32(4):P4.68, December 2025.One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be obtained by considering the maximum integer $k$ such that $H$ contains $k$ vertex-disjoint cycles. There exists a graph of treewidth ${Ω(k\log k)}$ which does not contain $k$ vertex-disjoint cycles, from which it follows that ${f(H) = Ω(k\log k)}$. In particular, if ${f(H)}$ is linear in ${\lvert{V(H)}\rvert}$ for graphs $H$ from a subclass of planar graphs, it is necessary that $n$-vertex graphs from the class contain at most ${O(n/\log(n))}$ vertex-disjoint cycles. We ask whether this is also a sufficient condition, and demonstrate that this is true for classes of planar graphs with bounded component size. For an $n$-vertex graph $H$ which is a disjoint union of $r$ cycles, we show that ${f(H) \leq 3n/2 + O(r^2 \log r)}$, and improve this to ${f(H) \leq n + O(\sqrt{n})}$ when ${r = 2}$. In particular this bound is linear when ${r=O(\sqrt{n}/\log(n))}$. We present a linear bound for ${f(H)}$ when $H$ is a subdivision of an $r$-edge planar graph for any constant $r$. We also improve the best known bounds for ${f(H)}$ when $H$ is the wheel graph or the ${4 \times 4}$ grid, obtaining a bound of $160$ for the latter.
Linear bounds on treewidth in terms of excluded planar minors
One of the fundamental results in graph minor theory is that for every planar graph $H$, there is a minimum integer $f(H)$ such that graphs with no minor isomorphic to $H$ have treewidth at most $f(H)$. A lower bound for ${f(H)}$ can be obtained by considering the maximum integer $k$ such that $H$ contains $k$ vertex-disjoint cycles. There exists a graph of treewidth ${Ω(k\log k)}$ which does not contain $k$ vertex-disjoint cycles, from which it follows that ${f(H) = Ω(k\log k)}$. In particular, if ${f(H)}$ is linear in ${\lvert{V(H)}\rvert}$ for graphs $H$ from a subclass of planar graphs, it is necessary that $n$-vertex graphs from the class contain at most ${O(n/\log(n))}$ vertex-disjoint cycles. We ask whether this is also a sufficient condition, and demonstrate that this is true for classes of planar graphs with bounded component size. For an $n$-vertex graph $H$ which is a disjoint union of $r$ cycles, we show that ${f(H) \leq 3n/2 + O(r^2 \log r)}$, and improve this to ${f(H) \leq n + O(\sqrt{n})}$ when ${r = 2}$. In particular this bound is linear when ${r=O(\sqrt{n}/\log(n))}$. We present a linear bound for ${f(H)}$ when $H$ is a subdivision of an $r$-edge planar graph for any constant $r$. We also improve the best known bounds for ${f(H)}$ when $H$ is the wheel graph or the ${4 \times 4}$ grid, obtaining a bound of $160$ for the latter.
- The proper conflict-free k-coloring problem and the odd k-coloring problem are NP-complete on bipartite graphs (with Jungho Ahn안정호 and Seonghyuk Im임성혁), Discrete Appl. Math., 377:10-17, December 2025.A proper coloring of a graph is \emph{proper conflict-free} if every non-isolated vertex $v$ has a neighbor whose color is unique in the neighborhood of $v$. A proper coloring of a graph is \emph{odd} if for every non-isolated vertex $v$, there is a color appearing an odd number of times in the neighborhood of $v$. For an integer $k$, the \textsc{PCF $k$-Coloring} problem asks whether an input graph admits a proper conflict-free $k$-coloring and the \textsc{Odd $k$-Coloring} asks whether an input graph admits an odd $k$-coloring. We show that for every integer $k\geq3$, both problems are NP-complete, even if the input graph is bipartite. Furthermore, we show that the \textsc{PCF $4$-Coloring} problem is NP-complete when the input graph is planar.
The proper conflict-free k-coloring problem and the odd k-coloring problem are NP-complete on bipartite graphs
A proper coloring of a graph is \emph{proper conflict-free} if every non-isolated vertex $v$ has a neighbor whose color is unique in the neighborhood of $v$. A proper coloring of a graph is \emph{odd} if for every non-isolated vertex $v$, there is a color appearing an odd number of times in the neighborhood of $v$. For an integer $k$, the \textsc{PCF $k$-Coloring} problem asks whether an input graph admits a proper conflict-free $k$-coloring and the \textsc{Odd $k$-Coloring} asks whether an input graph admits an odd $k$-coloring. We show that for every integer $k\geq3$, both problems are NP-complete, even if the input graph is bipartite. Furthermore, we show that the \textsc{PCF $4$-Coloring} problem is NP-complete when the input graph is planar.
- A unified Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups (with J. Pascal Gollin, Kevin Hendrey, O-joung Kwon권오정, and Youngho Yoo유영호), Math. Ann., 393:2507-2559, October 2025.In 1965, Erdős and Pósa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs ${(\ell, z)}$ of integers where such a duality holds for the family of cycles of length $\ell$ modulo $z$. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group. This unifies almost all known types of cycles that admit such a duality, and it also provides new results. Moreover, we characterise the obstructions to such a duality in this setting, and thereby obtain an analogous characterisation for cycles in graphs embeddable on a fixed compact orientable surface.
A unified Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups
In 1965, Erdős and Pósa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter and Neumann-Lara asked in 1988 to find all pairs ${(\ell, z)}$ of integers where such a duality holds for the family of cycles of length $\ell$ modulo $z$. We characterise all such pairs, and we further generalise this characterisation to cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group. This unifies almost all known types of cycles that admit such a duality, and it also provides new results. Moreover, we characterise the obstructions to such a duality in this setting, and thereby obtain an analogous characterisation for cycles in graphs embeddable on a fixed compact orientable surface.
- Space-efficient parameterized algorithms on graphs of low shrubdepth (with Vera Chekan, Robert Ganian, Mamadou Moustapha Kanté, Michał Pilipczuk, Erik Jan van Leeuwen, Benjamin Bergougnoux, and Matthias Mnich), ACM Trans. Comput. Theory, 17(3):1-42, September 2025.Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on $n$-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth $d$ and using $k$ labels, we can solve - Independent Set in time $2^{O(dk)}\cdot n^{O(1)}$ using $O(dk^2\log n)$ space; - Max Cut in time $n^{O(dk)}$ using $O(dk\log n)$ space; and - Dominating Set in time $2^{O(dk)}\cdot n^{O(1)}$ using $n^{O(1)}$ space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with $d$ if one wishes to keep the space complexity polynomial.
Space-efficient parameterized algorithms on graphs of low shrubdepth
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on $n$-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth $d$ and using $k$ labels, we can solve - Independent Set in time $2^{O(dk)}\cdot n^{O(1)}$ using $O(dk^2\log n)$ space; - Max Cut in time $n^{O(dk)}$ using $O(dk\log n)$ space; and - Dominating Set in time $2^{O(dk)}\cdot n^{O(1)}$ using $n^{O(1)}$ space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with $d$ if one wishes to keep the space complexity polynomial.
- Note on Hamiltonicity of basis graphs of even delta-matroids (with Donggyu Kim김동규), J. Graph Theory, 109(4):446-453, August 2025.We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two distinct edges $e$ and $f$ sharing a common end, it has a Hamiltonian cycle using $e$ and avoiding $f$ unless it has at most two vertices or it is a cycle of length at most four. We also prove that if the basis graph is not a hypercube graph, then each vertex belongs to cycles of every length $\ell\ge 3$, and each edge belongs to cycles of every length $\ell \ge 4$. For the last theorem, we provide two proofs, one of which uses the result of Naddef (1984) on polytopes and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the other is a direct proof using various properties of even delta-matroids. Our theorems generalize the analogous results for matroids by Holzmann and Harary (1972) and Bondy and Ingleton (1976).
Note on Hamiltonicity of basis graphs of even delta-matroids
We show that the basis graph of an even delta-matroid is Hamiltonian if it has more than two vertices. More strongly, we prove that for two distinct edges $e$ and $f$ sharing a common end, it has a Hamiltonian cycle using $e$ and avoiding $f$ unless it has at most two vertices or it is a cycle of length at most four. We also prove that if the basis graph is not a hypercube graph, then each vertex belongs to cycles of every length $\ell\ge 3$, and each edge belongs to cycles of every length $\ell \ge 4$. For the last theorem, we provide two proofs, one of which uses the result of Naddef (1984) on polytopes and the result of Chepoi (2007) on basis graphs of even delta-matroids, and the other is a direct proof using various properties of even delta-matroids. Our theorems generalize the analogous results for matroids by Holzmann and Harary (1972) and Bondy and Ingleton (1976).
- Clustered colouring of odd-H-minor-free graphs (with Robert Hickingbotham, Dong Yeap Kang강동엽, Raphael Steiner, and David R. Wood), In: D. R. Wood, A. M. Etheridge, J. de Gier, N. Joshi (eds), 2023 MATRIX Annals. MATRIX Book Series, vol 6., pp. 73-80, Springer, July 2025.The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $c$ such that every graph $G\in\mathcal{G}$ has a $c$-colouring where each monochromatic component in $G$ has bounded size. We study the clustered chromatic number of graph classes $\mathcal{G}_H^{\text{odd}}$ defined by excluding a graph $H$ as an odd-minor. How does the structure of $H$ relate to the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$? We adapt a proof method of Norin, Scott, Seymour and Wood (2019) to show that the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$ is tied to the tree-depth of $H$.
Clustered colouring of odd-H-minor-free graphs
The clustered chromatic number of a graph class $\mathcal{G}$ is the minimum integer $c$ such that every graph $G\in\mathcal{G}$ has a $c$-colouring where each monochromatic component in $G$ has bounded size. We study the clustered chromatic number of graph classes $\mathcal{G}_H^{\text{odd}}$ defined by excluding a graph $H$ as an odd-minor. How does the structure of $H$ relate to the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$? We adapt a proof method of Norin, Scott, Seymour and Wood (2019) to show that the clustered chromatic number of $\mathcal{G}_H^{\text{odd}}$ is tied to the tree-depth of $H$.
- Twin-width of subdivisions of multigraphs (with Jungho Ahn안정호, Debsoumya Chakraborti, and Kevin Hendrey), SIAM J. Discrete Math., 39(2):607-862, April 2025.For each $d\leq3$, we construct a finite set $F_d$ of multigraphs such that for each graph $H$ of girth at least $5$ obtained from a multigraph $G$ by subdividing each edge at least two times, $H$ has twin-width at most $d$ if and only if $G$ has no minor in $F_d$. This answers a question of Bergé, Bonnet, and Déprés asking for the structure of graphs $G$ such that each long subdivision of $G$ has twin-width $4$. As a corollary, we show that the $7\times7$ grid has twin-width $4$, which answers a question of Schidler and Szeider.
Twin-width of subdivisions of multigraphs
For each $d\leq3$, we construct a finite set $F_d$ of multigraphs such that for each graph $H$ of girth at least $5$ obtained from a multigraph $G$ by subdividing each edge at least two times, $H$ has twin-width at most $d$ if and only if $G$ has no minor in $F_d$. This answers a question of Bergé, Bonnet, and Déprés asking for the structure of graphs $G$ such that each long subdivision of $G$ has twin-width $4$. As a corollary, we show that the $7\times7$ grid has twin-width $4$, which answers a question of Schidler and Szeider.
2024
- Twin-width of random graphs (with Jungho Ahn안정호, Debsoumya Chakraborti, Kevin Hendrey, and Donggyu Kim김동규), Random Structures Algorithms, 65(4):794-831, December 2024.We investigate the twin-width of the Erdős-Rényi random graph $G(n,p)$. We unveil a surprising behavior of this parameter by showing the existence of a constant $p^*\approx 0.4$ such that with high probability, when $p^*\le p\le 1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $0<p<p^*$ or $1>p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In addition, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2 - \sqrt{3n \log n}/2$ within an interval of length $o(\sqrt{n\log n})$. For the sparse random graph, we show that with high probability, the twin-width of $G(n,p)$ is $Θ(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$.
Twin-width of random graphs
We investigate the twin-width of the Erdős-Rényi random graph $G(n,p)$. We unveil a surprising behavior of this parameter by showing the existence of a constant $p^*\approx 0.4$ such that with high probability, when $p^*\le p\le 1-p^*$, the twin-width is asymptotically $2p(1-p)n$, whereas, when $0<p<p^*$ or $1>p>1-p^*$, the twin-width is significantly higher than $2p(1-p)n$. In addition, we show that the twin-width of $G(n,1/2)$ is concentrated around $n/2 - \sqrt{3n \log n}/2$ within an interval of length $o(\sqrt{n\log n})$. For the sparse random graph, we show that with high probability, the twin-width of $G(n,p)$ is $Θ(n\sqrt{p})$ when $(726\ln n)/n\leq p\leq1/2$.
- Vertex-minors of graphs: A survey (with Donggyu Kim김동규), Discrete Appl. Math., 351:54-73, July 2024.For a vertex v of a graph, the local complementation at v is an operation that replaces the neighborhood of v by its complement graph. Two graphs are locally equivalent if one is obtained from the other by a sequence of local complementations. A graph H is a vertex- minor of a graph G if H is an induced subgraph of a graph locally equivalent to G. Although this concept was introduced in the 1980s, it was not widely known and except for the survey paper of Bouchet published in 1990, there is no comprehensive survey listing all the new developments. We survey classic and recent theorems and conjectures on vertex-minors and related concepts such as circle graphs, cut-rank functions, rank-width, and interlace polynomials.
Vertex-minors of graphs: A survey
For a vertex v of a graph, the local complementation at v is an operation that replaces the neighborhood of v by its complement graph. Two graphs are locally equivalent if one is obtained from the other by a sequence of local complementations. A graph H is a vertex- minor of a graph G if H is an induced subgraph of a graph locally equivalent to G. Although this concept was introduced in the 1980s, it was not widely known and except for the survey paper of Bouchet published in 1990, there is no comprehensive survey listing all the new developments. We survey classic and recent theorems and conjectures on vertex-minors and related concepts such as circle graphs, cut-rank functions, rank-width, and interlace polynomials.
Source: local-pdf:/legacy/2024/03/2023vertexminors-survey-revised.pdf
- Prime vertex-minors of a prime graph (with Donggyu Kim김동규), European J. Combin., 118:103871, May 2024.A graph is prime if it does not admit a partition $(A,B)$ of its vertex set such that $\min\{|A|,|B|\} \geq 2$ and the rank of the $A\times B$ submatrix of its adjacency matrix is at most $1$. A vertex $v$ of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at $v$ result in prime graphs. In 1994, Allys proved that every prime graph with at least four vertices has a non-essential vertex unless it is locally equivalent to a cycle graph. We prove that every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle graph. As a corollary, we show that for a prime graph $G$ with at least six vertices and a vertex $x$, there is a vertex $v \ne x$ such that $G \setminus v$ or $G * v \setminus v$ is prime, unless $x$ is adjacent to all other vertices and $G$ is isomorphic to a particular graph on odd number of vertices. Furthermore, we show that a prime graph with at least four vertices has at least three non-essential vertices, unless it is locally equivalent to a graph consisting of at least two internally-disjoint paths between two fixed distinct vertices having no common neighbors. We also prove analogous results for pivot-minors.
Prime vertex-minors of a prime graph
A graph is prime if it does not admit a partition $(A,B)$ of its vertex set such that $\min\{|A|,|B|\} \geq 2$ and the rank of the $A\times B$ submatrix of its adjacency matrix is at most $1$. A vertex $v$ of a graph is non-essential if at least two of the three kinds of vertex-minor reductions at $v$ result in prime graphs. In 1994, Allys proved that every prime graph with at least four vertices has a non-essential vertex unless it is locally equivalent to a cycle graph. We prove that every prime graph with at least four vertices has at least two non-essential vertices unless it is locally equivalent to a cycle graph. As a corollary, we show that for a prime graph $G$ with at least six vertices and a vertex $x$, there is a vertex $v \ne x$ such that $G \setminus v$ or $G * v \setminus v$ is prime, unless $x$ is adjacent to all other vertices and $G$ is isomorphic to a particular graph on odd number of vertices. Furthermore, we show that a prime graph with at least four vertices has at least three non-essential vertices, unless it is locally equivalent to a graph consisting of at least two internally-disjoint paths between two fixed distinct vertices having no common neighbors. We also prove analogous results for pivot-minors.
- A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups (with J. Pascal Gollin, Kevin Hendrey, Ken-ichi Kawarabayashi河原林 健一, and O-joung Kwon권오정), J. Lon. Math. Soc., 109(1):e12858, January 2024.Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example cycles of length at least $\ell$, cycles of length $p$ modulo $q$, cycles intersecting a prescribed set of vertices at least $t$ times, and cycles contained in given $\mathbb{Z}_2$-homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.
A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups
Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles. A multitude of natural properties of cycles can be encoded in this setting, for example cycles of length at least $\ell$, cycles of length $p$ modulo $q$, cycles intersecting a prescribed set of vertices at least $t$ times, and cycles contained in given $\mathbb{Z}_2$-homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.
2023
- A chain theorem for sequentially 3-rank-connected graphs with respect to vertex-minors (with Duksang Lee이덕상), European J. Combin., 113:103761, October 2023.Tutte (1961) proved the chain theorem for simple $3$-connected graphs with respect to minors, which states that every simple $3$-connected graph $G$ has a simple $3$-connected minor with one edge fewer than $G$, unless $G$ is a wheel graph. Bouchet (1987) proved an analog for prime graphs with respect to vertex-minors. We present a chain theorem for higher connectivity with respect to vertex-minors, showing that every sequentially $3$-rank-connected graph $G$ has a sequentially $3$-rank-connected vertex-minor with one vertex fewer than $G$, unless $|V(G)|\leq 12$.
A chain theorem for sequentially 3-rank-connected graphs with respect to vertex-minors
Tutte (1961) proved the chain theorem for simple $3$-connected graphs with respect to minors, which states that every simple $3$-connected graph $G$ has a simple $3$-connected minor with one edge fewer than $G$, unless $G$ is a wheel graph. Bouchet (1987) proved an analog for prime graphs with respect to vertex-minors. We present a chain theorem for higher connectivity with respect to vertex-minors, showing that every sequentially $3$-rank-connected graph $G$ has a sequentially $3$-rank-connected vertex-minor with one vertex fewer than $G$, unless $|V(G)|\leq 12$.
- A polynomial kernel for 3-leaf power deletion (with Jungho Ahn안정호, Eduard Eiben, and O-joung Kwon권오정), Algorithmica, 85(10):3058-3087, October 2023. (An extended abstract appeared in MFCS 2020.).For a non-negative integer $\ell$, the $\ell$-leaf power of a tree $T$ is a simple graph $G$ on the leaves of $T$ such that two vertices are adjacent in $G$ if and only if their distance in $T$ is at most $\ell$. We provide a polynomial kernel for the problem of deciding whether we can delete at most $k$ vertices to make an input graph a $3$-leaf power of some tree. More specifically, we present a polynomial-time algorithm for an input instance $(G,k)$ for the problem to output an equivalent instance $(G',k')$ such that $k'\leq k$ and $G'$ has at most $O(k^{14})$ vertices.
A polynomial kernel for 3-leaf power deletion
For a non-negative integer $\ell$, the $\ell$-leaf power of a tree $T$ is a simple graph $G$ on the leaves of $T$ such that two vertices are adjacent in $G$ if and only if their distance in $T$ is at most $\ell$. We provide a polynomial kernel for the problem of deciding whether we can delete at most $k$ vertices to make an input graph a $3$-leaf power of some tree. More specifically, we present a polynomial-time algorithm for an input instance $(G,k)$ for the problem to output an equivalent instance $(G',k')$ such that $k'\leq k$ and $G'$ has at most $O(k^{14})$ vertices.
- Γ-graphic delta-matroids and their applications (with Donggyu Kim김동규 and Duksang Lee이덕상), Combinatorica, 43(5):963-983, October 2023.For an abelian group $Γ$, a $Γ$-labelled graph is a graph whose vertices are labelled by elements of $Γ$. We prove that a certain collection of edge sets of a $Γ$-labelled graph forms a delta-matroid, which we call a $Γ$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by $k$ and Maximum Weight $S$-Tree Packing. We also discuss various properties of $Γ$-graphic delta-matroids.
Γ-graphic delta-matroids and their applications
For an abelian group $Γ$, a $Γ$-labelled graph is a graph whose vertices are labelled by elements of $Γ$. We prove that a certain collection of edge sets of a $Γ$-labelled graph forms a delta-matroid, which we call a $Γ$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by $k$ and Maximum Weight $S$-Tree Packing. We also discuss various properties of $Γ$-graphic delta-matroids.
- Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k (with Mamadou Moustapha Kanté, Eun Jung Kim김은정, and O-joung Kwon권오정), J. Combin. Theory Ser. B, 160:15-35, May 2023.Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field $\mathbb F$, the list needs to contain only finitely many $\mathbb F$-representable matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these $\mathbb F$-representable excluded minors in general. We consider the class of matroids of path-width at most $k$ for fixed $k$. We prove that for a finite field $\mathbb F$, every $\mathbb F$-representable excluded minor for the class of matroids of path-width at most $k$ has at most $2^{|\mathbb{F}|^{O(k^2)}}$ elements. We can therefore compute, for any integer $k$ and a fixed finite field $\mathbb F$, the set of $\mathbb F$-representable excluded minors for the class of matroids of path-width $k$, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an $\mathbb F$-represented matroid is at most $k$. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most $k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.
Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k
Every minor-closed class of matroids of bounded branch-width can be characterized by a list of excluded minors, but unlike graphs, this list may need to be infinite in general. However, for each fixed finite field $\mathbb F$, the list needs to contain only finitely many $\mathbb F$-representable matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these $\mathbb F$-representable excluded minors in general. We consider the class of matroids of path-width at most $k$ for fixed $k$. We prove that for a finite field $\mathbb F$, every $\mathbb F$-representable excluded minor for the class of matroids of path-width at most $k$ has at most $2^{|\mathbb{F}|^{O(k^2)}}$ elements. We can therefore compute, for any integer $k$ and a fixed finite field $\mathbb F$, the set of $\mathbb F$-representable excluded minors for the class of matroids of path-width $k$, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an $\mathbb F$-represented matroid is at most $k$. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most $k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.
- Rank connectivity and pivot-minors of graphs, European J. Combin., 108:103634, February 2023.The cut-rank of a set $X$ in a graph $G$ is the rank of the $X\times (V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$ is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is prime (with respect to the split decomposition) if it is connected and has no splits. A graph $G$ is $k^{+\ell}$-rank-connected if for every set $X$ of vertices with the cut-rank less than $k$, $\lvert X\rvert$ or $\lvert V(G)-X\rvert $ is less than $k+\ell$. We prove that every prime $3^{+2}$-rank-connected graph $G$ with at least $10$ vertices has a prime $3^{+3}$-rank-connected pivot-minor $H$ such that $\lvert V(H)\rvert =\lvert V(G)\rvert -1$. As a corollary, we show that every excluded pivot-minor for the class of graphs of rank-width at most $k$ has at most $(3.5 \cdot 6^{k}-1)/5$ vertices for $k\ge 2$. We also show that the excluded pivot-minors for the class of graphs of rank-width at most $2$ have at most $16$ vertices.
Rank connectivity and pivot-minors of graphs
The cut-rank of a set $X$ in a graph $G$ is the rank of the $X\times (V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$ is less than $2$ and both $X$ and $Y$ have at least two vertices. A graph is prime (with respect to the split decomposition) if it is connected and has no splits. A graph $G$ is $k^{+\ell}$-rank-connected if for every set $X$ of vertices with the cut-rank less than $k$, $\lvert X\rvert$ or $\lvert V(G)-X\rvert $ is less than $k+\ell$. We prove that every prime $3^{+2}$-rank-connected graph $G$ with at least $10$ vertices has a prime $3^{+3}$-rank-connected pivot-minor $H$ such that $\lvert V(H)\rvert =\lvert V(G)\rvert -1$. As a corollary, we show that every excluded pivot-minor for the class of graphs of rank-width at most $k$ has at most $(3.5 \cdot 6^{k}-1)/5$ vertices for $k\ge 2$. We also show that the excluded pivot-minors for the class of graphs of rank-width at most $2$ have at most $16$ vertices.
- Independent domination of graphs with bounded maximum degree (with Eun-Kyung Cho조은경, Jinha Kim김진하, and Minki Kim김민기), J. Combin. Theory Ser. B, 158:341-352, January 2023.An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $Δ=4$ or $Δ\ge 6$, every connected $n$-vertex graph of maximum degree at most $Δ$ has an independent dominating set of size at most $(1-\fracΔ{\lfloorΔ^2/4\rfloor+Δ})(n-1)+1$. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most $(1-\fracΔ{ \lfloorΔ^2/4\rfloor+Δ})n$.
Independent domination of graphs with bounded maximum degree
An independent dominating set of a graph, also known as a maximal independent set, is a set $S$ of pairwise non-adjacent vertices such that every vertex not in $S$ is adjacent to some vertex in $S$. We prove that for $Δ=4$ or $Δ\ge 6$, every connected $n$-vertex graph of maximum degree at most $Δ$ has an independent dominating set of size at most $(1-\fracΔ{\lfloorΔ^2/4\rfloor+Δ})(n-1)+1$. In addition, we characterize all connected graphs having the equality and we show that other connected graphs have an independent dominating set of size at most $(1-\fracΔ{ \lfloorΔ^2/4\rfloor+Δ})n$.
- Intertwining connectivities for vertex-minors and pivot-minors (with Duksang Lee이덕상), SIAM J. Discrete Math., 37(1):304-314, 2023.We show that for pairs $(Q,R)$ and $(S,T)$ of disjoint subsets of vertices of a graph $G$, if $G$ is sufficiently large, then there exists a vertex $v$ in $V(G)-(Q\cup R\cup S\cup T)$ such that there are two ways to reduce $G$ by a vertex-minor operation that removes $v$ while preserving the connectivity between $Q$ and $R$ and the connectivity between $S$ and $T$. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids.
Intertwining connectivities for vertex-minors and pivot-minors
We show that for pairs $(Q,R)$ and $(S,T)$ of disjoint subsets of vertices of a graph $G$, if $G$ is sufficiently large, then there exists a vertex $v$ in $V(G)-(Q\cup R\cup S\cup T)$ such that there are two ways to reduce $G$ by a vertex-minor operation that removes $v$ while preserving the connectivity between $Q$ and $R$ and the connectivity between $S$ and $T$. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids.
2022
- Obstructions for partitioning into forests and outerplanar graphs (with Ringi Kim김린기 and Sergey Norin), Discrete Appl. Math., 312:15-28, May 2022. [open access].For a class $\mathcal C$ of graphs, we define $\mathcal C$-edge-brittleness of a graph $G$ as the minimum $\ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ edges having ends in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-edge-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of graphs with no $K_4\setminus e$ topological minors in terms of forbidden obstructions. We also define $\mathcal C$-vertex-brittleness of a graph $G$ as the minimum $\ell$ such that the edge set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ vertices incident with edges in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-vertex-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of outerplanar graphs in terms of forbidden obstructions. We also investigate the relations between the new parameters and the edit distance.
Obstructions for partitioning into forests and outerplanar graphs
For a class $\mathcal C$ of graphs, we define $\mathcal C$-edge-brittleness of a graph $G$ as the minimum $\ell$ such that the vertex set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ edges having ends in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-edge-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of graphs with no $K_4\setminus e$ topological minors in terms of forbidden obstructions. We also define $\mathcal C$-vertex-brittleness of a graph $G$ as the minimum $\ell$ such that the edge set of $G$ can be partitioned into sets inducing a subgraph in $\mathcal C$ and there are $\ell$ vertices incident with edges in distinct parts. We characterize classes of graphs having bounded $\mathcal C$-vertex-brittleness for a class $\mathcal C$ of forests or a class $\mathcal C$ of outerplanar graphs in terms of forbidden obstructions. We also investigate the relations between the new parameters and the edit distance.
- Characterizing matroids whose bases form graphic delta-matroids (with Duksang Lee이덕상), European J. Combin., 101:103476, March 2022.We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most $48$ elements.
Characterizing matroids whose bases form graphic delta-matroids
We introduce delta-graphic matroids, which are matroids whose bases form graphic delta-matroids. The class of delta-graphic matroids contains graphic matroids as well as cographic matroids and is a proper subclass of the class of regular matroids. We give a structural characterization of the class of delta-graphic matroids. We also show that every forbidden minor for the class of delta-graphic matroids has at most $48$ elements.
- 3-degenerate induced subgraphs of a planar graph (with H. A. Kierstead, Xuding Zhu, Yangyan Gu, and Hao Qi), J. Graph Theory, 99(2):251-277, February 2022.A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$.
3-degenerate induced subgraphs of a planar graph
A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$.
- Bounds for the twin-width of graphs (with Jungho Ahn안정호, Kevin Hendrey, and Donggyu Kim김동규), SIAM J. Discrete Math., 36(3):2352-2366, 2022.Bonnet, Kim, Thomassé, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdős-Rényi random graph $G(n,p)$ with $1/n\leq p=p(n)\leq 1/2$ is larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for any positive $\varepsilon$. Lastly, we calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$.
Bounds for the twin-width of graphs
Bonnet, Kim, Thomassé, and Watrigant (2020) introduced the twin-width of a graph. We show that the twin-width of an $n$-vertex graph is less than $(n+\sqrt{n\ln n}+\sqrt{n}+2\ln n)/2$, and the twin-width of an $m$-edge graph for a positive $m$ is less than $\sqrt{3m}+ m^{1/4} \sqrt{\ln m} / (4\cdot 3^{1/4}) + 3m^{1/4} / 2$. Conference graphs of order $n$ (when such graphs exist) have twin-width at least $(n-1)/2$, and we show that Paley graphs achieve this lower bound. We also show that the twin-width of the Erdős-Rényi random graph $G(n,p)$ with $1/n\leq p=p(n)\leq 1/2$ is larger than $2p(1-p)n - (2\sqrt{2}+\varepsilon)\sqrt{p(1-p)n\ln n}$ asymptotically almost surely for any positive $\varepsilon$. Lastly, we calculate the twin-width of random graphs $G(n,p)$ with $p\leq c/n$ for a constant $c<1$, determining the thresholds at which the twin-width jumps from $0$ to $1$ and from $1$ to $2$.
2021
- Tree pivot-minors and linear rank-width (with Konrad K. Dabrowski, François Dross, Mamadou Moustapha Kanté, O-joung Kwon권오정, Daniël Paulusma, and Jisu Jeong정지수), SIAM J. Discrete Math., 35(4), 2922-2945, December 2021.Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) proved that for every tree~$T$, the class of graphs that do not contain $T$ as a minor has bounded path-width. For the pivot-minor relation, rank-width and linear rank-width take over the role from tree-width and path-width. As such, it is natural to examine if for every tree~$T$, the class of graphs that do not contain $T$ as a pivot-minor has bounded linear rank-width. We first prove that this statement is false whenever $T$ is a tree that is not a caterpillar. We conjecture that the statement is true if $T$ is a caterpillar. We are also able to give partial confirmation of this conjecture by proving: (1) for every tree $T$, the class of $T$-pivot-minor-free distance-hereditary graphs has bounded linear rank-width if and only if $T$ is a caterpillar; (2) for every caterpillar $T$ on at most four vertices, the class of $T$-pivot-minor-free graphs has bounded linear rank-width. To prove our second result, we only need to consider $T=P_4$ and $T=K_{1,3}$, but we follow a general strategy: first we show that the class of $T$-pivot-minor-free graphs is contained in some class of $(H_1,H_2)$-free graphs, which we then show to have bounded linear rank-width. In particular, we prove that the class of $(K_3,S_{1,2,2})$-free graphs has bounded linear rank-width, which strengthens a known result that this graph class has bounded rank-width.
Tree pivot-minors and linear rank-width
Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) proved that for every tree~$T$, the class of graphs that do not contain $T$ as a minor has bounded path-width. For the pivot-minor relation, rank-width and linear rank-width take over the role from tree-width and path-width. As such, it is natural to examine if for every tree~$T$, the class of graphs that do not contain $T$ as a pivot-minor has bounded linear rank-width. We first prove that this statement is false whenever $T$ is a tree that is not a caterpillar. We conjecture that the statement is true if $T$ is a caterpillar. We are also able to give partial confirmation of this conjecture by proving: (1) for every tree $T$, the class of $T$-pivot-minor-free distance-hereditary graphs has bounded linear rank-width if and only if $T$ is a caterpillar; (2) for every caterpillar $T$ on at most four vertices, the class of $T$-pivot-minor-free graphs has bounded linear rank-width. To prove our second result, we only need to consider $T=P_4$ and $T=K_{1,3}$, but we follow a general strategy: first we show that the class of $T$-pivot-minor-free graphs is contained in some class of $(H_1,H_2)$-free graphs, which we then show to have bounded linear rank-width. In particular, we prove that the class of $(K_3,S_{1,2,2})$-free graphs has bounded linear rank-width, which strengthens a known result that this graph class has bounded rank-width.
- Finding branch-decompositions of matroids, hypergraphs, and more (with Jisu Jeong정지수 and Eun Jung Kim김은정), SIAM J. Discrete Math., 35(4):2544-2617, November 2021. (An extended abstract appeared in ICALP 2018.).Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathbb F$, we wish to find a "branch-decomposition" of these subspaces of width at most $k$ that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated to the leaves in one component of $T-e$ and the sum of subspaces associated to the leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathbb F$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $\mathbb F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of $\mathbb F$-represented matroids was due to Hliněný and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathbb F$-represented matroid. But their method is highly indirect. Their algorithm uses the nontrivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hliněný (2006) on checking monadic second-order formulas on $\mathbb F$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.
Finding branch-decompositions of matroids, hypergraphs, and more
Given $n$ subspaces of a finite-dimensional vector space over a fixed finite field $\mathbb F$, we wish to find a "branch-decomposition" of these subspaces of width at most $k$ that is a subcubic tree $T$ with $n$ leaves mapped bijectively to the subspaces such that for every edge $e$ of $T$, the sum of subspaces associated to the leaves in one component of $T-e$ and the sum of subspaces associated to the leaves in the other component have the intersection of dimension at most $k$. This problem includes the problems of computing branch-width of $\mathbb F$-represented matroids, rank-width of graphs, branch-width of hypergraphs, and carving-width of graphs. We present a fixed-parameter algorithm to construct such a branch-decomposition of width at most $k$, if it exists, for input subspaces of a finite-dimensional vector space over $\mathbb F$. Our algorithm is analogous to the algorithm of Bodlaender and Kloks (1996) on tree-width of graphs. To extend their framework to branch-decompositions of vector spaces, we developed highly generic tools for branch-decompositions on vector spaces. The only known previous fixed-parameter algorithm for branch-width of $\mathbb F$-represented matroids was due to Hliněný and Oum (2008) that runs in time $O(n^3)$ where $n$ is the number of elements of the input $\mathbb F$-represented matroid. But their method is highly indirect. Their algorithm uses the nontrivial fact by Geelen et al. (2003) that the number of forbidden minors is finite and uses the algorithm of Hliněný (2006) on checking monadic second-order formulas on $\mathbb F$-represented matroids of small branch-width. Our result does not depend on such a fact and is completely self-contained, and yet matches their asymptotic running time for each fixed $k$.
- Obstructions for bounded shrub-depth and rank-depth (with O-joung Kwon권오정, Rose McCarty, and Paul Wollan), J. Combin. Theory Ser. B, 149, 76-91, July 2021.Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hliněný, Kwon, Obdržálek, and Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertex-minor isomorphic to the path on $t$ vertices has bounded shrub-depth.
Obstructions for bounded shrub-depth and rank-depth
Shrub-depth and rank-depth are dense analogues of the tree-depth of a graph. It is well known that a graph has large tree-depth if and only if it has a long path as a subgraph. We prove an analogous statement for shrub-depth and rank-depth, which was conjectured by Hliněný, Kwon, Obdržálek, and Ordyniak [Tree-depth and vertex-minors, European J.~Combin. 2016]. Namely, we prove that a graph has large rank-depth if and only if it has a vertex-minor isomorphic to a long path. This implies that for every integer $t$, the class of graphs with no vertex-minor isomorphic to the path on $t$ vertices has bounded shrub-depth.
- Equitable partition of planar graphs (with Ringi Kim김린기 and Xin Zhang), Discrete Math., 344(6):112351, June 2021.An equitable $k$-partition of a graph $G$ is a collection of induced subgraphs $(G[V_1],G[V_2],\ldots,G[V_k])$ of $G$ such that $(V_1,V_2,\ldots,V_k)$ is a partition of $V(G)$ and $-1\le |V_i|-|V_j|\le 1$ for all $1\le i<j\le k$. We prove that every planar graph admits an equitable $2$-partition into $3$-degenerate graphs, an equitable $3$-partition into $2$-degenerate graphs, and an equitable $3$-partition into two forests and one graph.
Equitable partition of planar graphs
An equitable $k$-partition of a graph $G$ is a collection of induced subgraphs $(G[V_1],G[V_2],\ldots,G[V_k])$ of $G$ such that $(V_1,V_2,\ldots,V_k)$ is a partition of $V(G)$ and $-1\le |V_i|-|V_j|\le 1$ for all $1\le i<j\le k$. We prove that every planar graph admits an equitable $2$-partition into $3$-degenerate graphs, an equitable $3$-partition into $2$-degenerate graphs, and an equitable $3$-partition into two forests and one graph.
- Obstructions for bounded branch-depth in matroids (with J. Pascal Gollin, Kevin Hendrey, and Dillon Mayhew), Adv. Comb., 2021:4, 25pp, May 2021. [open access].DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.
Obstructions for bounded branch-depth in matroids
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.
- The Erdős-Hajnal property for graphs with no fixed cycle as a pivot-minor (with Jaehoon Kim김재훈), Electron. J. Combin., 28, #P2.9, April 2021.We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and $A$ is either complete or anticomplete to $B$. This proves the analog of the Erdős-Hajnal conjecture for the class of graphs with no pivot-minor isomorphic to $C_k$.
The Erdős-Hajnal property for graphs with no fixed cycle as a pivot-minor
We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and $A$ is either complete or anticomplete to $B$. This proves the analog of the Erdős-Hajnal conjecture for the class of graphs with no pivot-minor isomorphic to $C_k$.
- Graphs of bounded depth-2 rank-brittleness (with O-joung Kwon권오정), J. Graph Theory, 96, 361-378, March 2021.We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$ whose leaves are labelled by the vertices of a graph $G$, and the width is measured by the maximum possible cut-rank of a partition of $V(G)$ induced by splitting an internal node of the tree to make two components. The minimum width possible is called the depth-$2$ rank-brittleness of $G$. We prove that for all $n$, every graph with sufficiently large depth-$2$ rank-brittleness contains $P_n$ or disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ as a vertex-minor.
Graphs of bounded depth-2 rank-brittleness
We characterize classes of graphs closed under taking vertex-minors and having no $P_n$ and no disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ for some $n$. Our characterization is described in terms of a tree of radius $2$ whose leaves are labelled by the vertices of a graph $G$, and the width is measured by the maximum possible cut-rank of a partition of $V(G)$ induced by splitting an internal node of the tree to make two components. The minimum width possible is called the depth-$2$ rank-brittleness of $G$. We prove that for all $n$, every graph with sufficiently large depth-$2$ rank-brittleness contains $P_n$ or disjoint union of $n$ copies of the $1$-subdivision of $K_{1,n}$ as a vertex-minor.
- Tangle-tree duality in abstract separation systems (with Reinhard Diestel), Adv. Math., 377, 107470, January 2021. (This is an updated version of the first half of a manuscript “Unifying duality theorems for width parameters in graphs and matroids. I. Weak and strong duality“.).We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite different. The theorem asserts a duality between the existence of high cohesiveness somewhere local and a global overall tree structure. We describe cohesive substructures in a unified way in the format of tangles: as orientations of low-order separations satisfying certain consistency axioms. These axioms can be expressed without reference to the underlying structure, such as a graph or matroid, but just in terms of the poset of the separations themselves. This makes it possible to identify tangles, and apply our tangle-tree duality theorem, in very diverse settings. Our result implies all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width or rank-width. It yields new, tangle-type, duality theorems for tree-width and path-width. It implies the existence of width parameters dual to cohesive substructures such as $k$-blocks, edge-tangles, or given subsets of tangles, for which no width duality theorems were previously known. Abstract separation systems can be found also in structures quite unlike graphs and matroids. For example, our theorem can be applied to image analysis by capturing the regions of an image as tangles of separations defined as natural partitions of its set of pixels. It can be applied in big data contexts by capturing clusters as tangles. It can be applied in the social sciences, e.g. by capturing as tangles the few typical mindsets of individuals found by a survey. It could also be applied in pure mathematics, e.g. to separations of compact manifolds.
Tangle-tree duality in abstract separation systems
We prove a general width duality theorem for combinatorial structures with well-defined notions of cohesion and separation. These might be graphs and matroids, but can be much more general or quite different. The theorem asserts a duality between the existence of high cohesiveness somewhere local and a global overall tree structure. We describe cohesive substructures in a unified way in the format of tangles: as orientations of low-order separations satisfying certain consistency axioms. These axioms can be expressed without reference to the underlying structure, such as a graph or matroid, but just in terms of the poset of the separations themselves. This makes it possible to identify tangles, and apply our tangle-tree duality theorem, in very diverse settings. Our result implies all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width or rank-width. It yields new, tangle-type, duality theorems for tree-width and path-width. It implies the existence of width parameters dual to cohesive substructures such as $k$-blocks, edge-tangles, or given subsets of tangles, for which no width duality theorems were previously known. Abstract separation systems can be found also in structures quite unlike graphs and matroids. For example, our theorem can be applied to image analysis by capturing the regions of an image as tangles of separations defined as natural partitions of its set of pixels. It can be applied in big data contexts by capturing clusters as tangles. It can be applied in the social sciences, e.g. by capturing as tangles the few typical mindsets of individuals found by a survey. It could also be applied in pure mathematics, e.g. to separations of compact manifolds.
2020
- Branch-depth: Generalizing tree-depth of graphs (with Matt DeVos and O-joung Kwon권오정), European J. Combin., 90, 103186, December 2020. [open access].We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A $ of $E$ we let $λ_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $ρ_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $λ_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $ρ_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.
Branch-depth: Generalizing tree-depth of graphs
We present a concept called the branch-depth of a connectivity function, that generalizes the tree-depth of graphs. Then we prove two theorems showing that this concept aligns closely with the notions of tree-depth and shrub-depth of graphs as follows. For a graph $G = (V,E)$ and a subset $A $ of $E$ we let $λ_G (A)$ be the number of vertices incident with an edge in $A$ and an edge in $E \setminus A$. For a subset $X$ of $V$, let $ρ_G(X)$ be the rank of the adjacency matrix between $X$ and $V \setminus X$ over the binary field. We prove that a class of graphs has bounded tree-depth if and only if the corresponding class of functions $λ_G$ has bounded branch-depth and similarly a class of graphs has bounded shrub-depth if and only if the corresponding class of functions $ρ_G$ has bounded branch-depth, which we call the rank-depth of graphs. Furthermore we investigate various potential generalizations of tree-depth to matroids and prove that matroids representable over a fixed finite field having no large circuits are well-quasi-ordered by the restriction.
- The average cut-rank of graphs (with Tung H. Nguyen), European J. Combin., 90, 103183, December 2020. [open access].The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real $α$, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) $α$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) $α$ for each real $α\ge0$. Finally, we describe explicitly all graphs of average cut-rank at most $3/2$ and determine up to $3/2$ all possible values that can be realized as the average cut-rank of some graph.
The average cut-rank of graphs
The cut-rank of a set $X$ of vertices in a graph $G$ is defined as the rank of the $ X \times (V(G)\setminus X)$ matrix over the binary field whose $(i,j)$-entry is $1$ if the vertex $i$ in $X$ is adjacent to the vertex $j$ in $V(G)\setminus X$ and $0$ otherwise. We introduce the graph parameter called the average cut-rank of a graph, defined as the expected value of the cut-rank of a random set of vertices. We show that this parameter does not increase when taking vertex-minors of graphs and a class of graphs has bounded average cut-rank if and only if it has bounded neighborhood diversity. This allows us to deduce that for each real $α$, the list of induced-subgraph-minimal graphs having average cut-rank larger than (or at least) $α$ is finite. We further refine this by providing an upper bound on the size of obstruction and a lower bound on the number of obstructions for average cut-rank at most (or smaller than) $α$ for each real $α\ge0$. Finally, we describe explicitly all graphs of average cut-rank at most $3/2$ and determine up to $3/2$ all possible values that can be realized as the average cut-rank of some graph.
- Scattered classes of graphs (with O-joung Kwon권오정), SIAM J. Discrete Math., 34, no. 1, pp. 972-999, March 2020.For a class $\mathcal C$ of graphs $G$ equipped with functions $f_G$ defined on subsets of $E(G)$ or $V(G)$, we say that $\mathcal{C}$ is $k$-scattered with respect to $f_G$ if there exists a constant $\ell$ such that for every graph $G\in \mathcal C$, the domain of $f_G$ can be partitioned into subsets of size at most $k$ so that the union of every collection of the subsets has $f_G$ value at most $\ell$. We present structural characterizations of graph classes that are $k$-scattered with respect to several graph connectivity functions. In particular, our theorem for cut-rank functions provides a rough structural characterization of graphs having no $mK_{1,n}$ vertex-minor, which allows us to prove that such graphs have bounded linear rank-width.
Scattered classes of graphs
For a class $\mathcal C$ of graphs $G$ equipped with functions $f_G$ defined on subsets of $E(G)$ or $V(G)$, we say that $\mathcal{C}$ is $k$-scattered with respect to $f_G$ if there exists a constant $\ell$ such that for every graph $G\in \mathcal C$, the domain of $f_G$ can be partitioned into subsets of size at most $k$ so that the union of every collection of the subsets has $f_G$ value at most $\ell$. We present structural characterizations of graph classes that are $k$-scattered with respect to several graph connectivity functions. In particular, our theorem for cut-rank functions provides a rough structural characterization of graphs having no $mK_{1,n}$ vertex-minor, which allows us to prove that such graphs have bounded linear rank-width.
- Classes of graphs with no long cycle as a vertex-minor are polynomially 𝜒-bounded (with Ringi Kim김린기, O-joung Kwon권오정, and Vaidy Sivaraman), J. Combin. Theory Ser. B, 140, pp. 372-386, January 2020.A class $\mathcal G$ of graphs is $χ$-bounded if there is a function $f$ such that for every graph $G\in \mathcal G$ and every induced subgraph $H$ of $G$, $χ(H)\le f(ω(H))$. In addition, we say that $\mathcal G$ is polynomially $χ$-bounded if $f$ can be taken as a polynomial function. We prove that for every integer $n\ge3$, there exists a polynomial $f$ such that $χ(G)\le f(ω(G))$ for all graphs with no vertex-minor isomorphic to the cycle graph $C_n$. To prove this, we show that if $\mathcal G$ is polynomially $χ$-bounded, then so is the closure of $\mathcal G$ under taking the $1$-join operation.
Classes of graphs with no long cycle as a vertex-minor are polynomially 𝜒-bounded
A class $\mathcal G$ of graphs is $χ$-bounded if there is a function $f$ such that for every graph $G\in \mathcal G$ and every induced subgraph $H$ of $G$, $χ(H)\le f(ω(H))$. In addition, we say that $\mathcal G$ is polynomially $χ$-bounded if $f$ can be taken as a polynomial function. We prove that for every integer $n\ge3$, there exists a polynomial $f$ such that $χ(G)\le f(ω(G))$ for all graphs with no vertex-minor isomorphic to the cycle graph $C_n$. To prove this, we show that if $\mathcal G$ is polynomially $χ$-bounded, then so is the closure of $\mathcal G$ under taking the $1$-join operation.
2019
- Online Ramsey theory for a triangle on F-free graphs (with Hojin Choi최호진, Ilkyoo Choi최일규, and Jisu Jeong정지수), J. Graph Theory, 92, no. 2, pp. 152-171, October 2019.Given a class $\mathcal{C}$ of graphs and a fixed graph $H$, the online Ramsey game for $H$ on $\mathcal C$ is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builder introduces a new edge with the constraint that the resulting graph must be in $\mathcal C$, and Painter colors the new edge either red or blue. Builder wins the game if Painter is forced to make a monochromatic copy of $H$ at some point in the game. Otherwise, Painter can avoid creating a monochromatic copy of $H$ forever, and we say Painter wins the game. We initiate the study of characterizing the graphs $F$ such that for a given graph $H$, Painter wins the online Ramsey game for $H$ on $F$-free graphs. We characterize all graphs $F$ such that Painter wins the online Ramsey game for $C_3$ on the class of $F$-free graphs, except when $F$ is one particular graph. We also show that Painter wins the online Ramsey game for $C_3$ on the class of $K_4$-minor-free graphs, extending a result by Grytczuk, Hałuszczak, and Kierstead.
Online Ramsey theory for a triangle on F-free graphs
Given a class $\mathcal{C}$ of graphs and a fixed graph $H$, the online Ramsey game for $H$ on $\mathcal C$ is a game between two players Builder and Painter as follows: an unbounded set of vertices is given as an initial state, and on each turn Builder introduces a new edge with the constraint that the resulting graph must be in $\mathcal C$, and Painter colors the new edge either red or blue. Builder wins the game if Painter is forced to make a monochromatic copy of $H$ at some point in the game. Otherwise, Painter can avoid creating a monochromatic copy of $H$ forever, and we say Painter wins the game. We initiate the study of characterizing the graphs $F$ such that for a given graph $H$, Painter wins the online Ramsey game for $H$ on $F$-free graphs. We characterize all graphs $F$ such that Painter wins the online Ramsey game for $C_3$ on the class of $F$-free graphs, except when $F$ is one particular graph. We also show that Painter wins the online Ramsey game for $C_3$ on the class of $K_4$-minor-free graphs, extending a result by Grytczuk, Hałuszczak, and Kierstead.
- Improper colouring of graphs with no odd clique minor (with Dong Yeap Kang강동엽), Combin. Probab. Comput., 28, no. 5, pp. 740-754, September 2019.As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $6t-9$ sets $V_1, \dots, V_{6t-9}$ such that each $V_i$ induces a subgraph of bounded maximum degree. Secondly, we prove that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $10t-13$ sets $V_1, \dots, V_{10t-13}$ such that each $V_i$ induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into $496t$ such sets.
Improper colouring of graphs with no odd clique minor
As a strengthening of Hadwiger's conjecture, Gerards and Seymour conjectured that every graph with no odd $K_t$ minor is $(t-1)$-colorable. We prove two weaker variants of this conjecture. Firstly, we show that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $6t-9$ sets $V_1, \dots, V_{6t-9}$ such that each $V_i$ induces a subgraph of bounded maximum degree. Secondly, we prove that for each $t \geq 2$, every graph with no odd $K_t$ minor has a partition of its vertex set into $10t-13$ sets $V_1, \dots, V_{10t-13}$ such that each $V_i$ induces a subgraph with components of bounded size. The second theorem improves a result of Kawarabayashi (2008), which states that the vertex set can be partitioned into $496t$ such sets.
- Tangle-tree duality: in graphs, matroids and beyond (with Reinhard Diestel), Combinatorica, 39, no. 4, pp. 879-910, August 2019. (This is an updated version of the second half of a manuscript “Unifying duality theorems for width parameters in graphs and matroids. I. Weak and strong duality“.).We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets. Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a duality theorem for tree-width. Our results can be used to derive short proofs of all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width and rank-width.
Tangle-tree duality: in graphs, matroids and beyond
We apply a recent duality theorem for tangles in abstract separation systems to derive tangle-type duality theorems for width-parameters in graphs and matroids. We further derive a duality theorem for the existence of clusters in large data sets. Our applications to graphs include new, tangle-type, duality theorems for tree-width, path-width, and tree-decompositions of small adhesion. Conversely, we show that carving width is dual to edge-tangles. For matroids we obtain a duality theorem for tree-width. Our results can be used to derive short proofs of all the classical duality theorems for width parameters in graph minor theory, such as path-width, tree-width, branch-width and rank-width.
- Defective colouring of graphs excluding a subgraph or minor (with Patrice Ossona de Mendez and David R. Wood), Combinatorica, 39(2), pp. 377-410, April 2019.Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no $K_{t+1}$-minor can be $t$-coloured so that each colour class induces a subgraph of bounded maximum degree. We prove a common generalisation of these theorems with a weaker assumption about excluded subgraphs. This result leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness (with relevance to the earth-moon problem), graphs with given stack- or queue-number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, and graphs excluding a complete bipartite graph as a topological minor.
Defective colouring of graphs excluding a subgraph or minor
Archdeacon (1987) proved that graphs embeddable on a fixed surface can be $3$-coloured so that each colour class induces a subgraph of bounded maximum degree. Edwards, Kang, Kim, Oum and Seymour (2015) proved that graphs with no $K_{t+1}$-minor can be $t$-coloured so that each colour class induces a subgraph of bounded maximum degree. We prove a common generalisation of these theorems with a weaker assumption about excluded subgraphs. This result leads to new defective colouring results for several graph classes, including graphs with linear crossing number, graphs with given thickness (with relevance to the earth-moon problem), graphs with given stack- or queue-number, linklessly or knotlessly embeddable graphs, graphs with given Colin de Verdière parameter, and graphs excluding a complete bipartite graph as a topological minor.
- Chi-boundedness of graph classes excluding wheel vertex-minors (with O-joung Kwon권오정, Paul Wollan, and Hojin Choi최호진), J. Combin. Theory Ser. B, 135, pp. 319-348, March 2019.A class of graphs is $χ$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $χ$-bounded. This generalizes several previous results; $χ$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.
Chi-boundedness of graph classes excluding wheel vertex-minors
A class of graphs is $χ$-bounded if there exists a function $f:\mathbb N\rightarrow \mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less than or equal to $f(q)$. We denote by $W_n$ the wheel graph on $n+1$ vertices. We show that the class of graphs having no vertex-minor isomorphic to $W_n$ is $χ$-bounded. This generalizes several previous results; $χ$-boundedness for circle graphs, for graphs having no $W_5$ vertex-minors, and for graphs having no fan vertex-minors.
- Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs (with Robert Brignall, Hojin Choi최호진, and Jisu Jeong정지수), Discrete Applied Math., 257, pp. 60-66, March 2019.A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs.
Deciding whether there are infinitely many prime graphs with forbidden induced subgraphs
A homogeneous set of a graph $G$ is a set $X$ of vertices such that $2\le \lvert X\rvert <\lvert V(G)\rvert$ and no vertex in $V(G)-X$ has both a neighbor and a non-neighbor in $X$. A graph is prime if it has no homogeneous set. We present an algorithm to decide whether a class of graphs given by a finite set of forbidden induced subgraphs contains infinitely many non-isomorphic prime graphs.
2018
- A remark on the paper “Properties of intersecting families of ordered sets” by O. Einstein (with Sounggun Wee위성군), Combinatorica, 38, no 5., pp. 1279-1284, October 2018.O. Einstein (2008) proved Bollobás-type theorems on intersecting families of ordered sets of finite sets and subspaces. Unfortunately, we report that the proof of a theorem on ordered sets of subspaces had a mistake. We prove two weaker variants.
A remark on the paper “Properties of intersecting families of ordered sets” by O. Einstein
O. Einstein (2008) proved Bollobás-type theorems on intersecting families of ordered sets of finite sets and subspaces. Unfortunately, we report that the proof of a theorem on ordered sets of subspaces had a mistake. We prove two weaker variants.
- An FPT 2-approximation for tree-cut decomposition (with Eun Jung Kim김은정, Christophe Paul, Ignasi Sau, and Dimitrios M. Thilikos), Algorithmica, 80, no. 1, pp. 116-135, January 2018.The tree-cut width of a graph is a graph parameter defined by Wollan [J. Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized 2-approximation algorithm for the computation of tree-cut width; for an input $n$-vertex graph $G$ and an integer $w$, our algorithm either confirms that the tree-cut width of $G$ is more than $w$ or returns a tree-cut decomposition of $G$ certifying that its tree-cut width is at most $2w$, in time $2^{O(w^2\log w)} \cdot n^2$. Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known.
An FPT 2-approximation for tree-cut decomposition
The tree-cut width of a graph is a graph parameter defined by Wollan [J. Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized 2-approximation algorithm for the computation of tree-cut width; for an input $n$-vertex graph $G$ and an integer $w$, our algorithm either confirms that the tree-cut width of $G$ is more than $w$ or returns a tree-cut decomposition of $G$ certifying that its tree-cut width is at most $2w$, in time $2^{O(w^2\log w)} \cdot n^2$. Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known.
- Partitioning H-minor free graphs into three subgraphs with no large components (with Chun-Hung Liu), J. Combin. Theory Ser. B, 128, pp. 114-133, January 2018.We prove that for every graph $H$, if a graph $G$ has no (odd) $H$ minor, then its vertex set $V(G)$ can be partitioned into three sets $X_1$, $X_2$, $X_3$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$H$ and the maximum degree of~$G$. This improves a previous result of Alon, Ding, Oporowski and Vertigan~(2003) stating that $V(G)$ can be partitioned into four such sets if $G$ has no $H$ minor. Our theorem generalizes a result of Esperet and Joret~(2014), who proved it for graphs embeddable on a fixed surface and asked whether it is true for graphs with no $H$ minor. As a corollary, we prove that for every positive integer $t$, if a graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $3t$ sets $X_1,\ldots,X_{3t}$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$t$. This corollary improves a result of Wood~(2010), which states that $V(G)$ can be partitioned into $\lceil 3.5t+2\rceil$ such sets.
Partitioning H-minor free graphs into three subgraphs with no large components
We prove that for every graph $H$, if a graph $G$ has no (odd) $H$ minor, then its vertex set $V(G)$ can be partitioned into three sets $X_1$, $X_2$, $X_3$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$H$ and the maximum degree of~$G$. This improves a previous result of Alon, Ding, Oporowski and Vertigan~(2003) stating that $V(G)$ can be partitioned into four such sets if $G$ has no $H$ minor. Our theorem generalizes a result of Esperet and Joret~(2014), who proved it for graphs embeddable on a fixed surface and asked whether it is true for graphs with no $H$ minor. As a corollary, we prove that for every positive integer $t$, if a graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $3t$ sets $X_1,\ldots,X_{3t}$ such that for each~$i$, the subgraph induced on $X_i$ has no component of size larger than a function of~$t$. This corollary improves a result of Wood~(2010), which states that $V(G)$ can be partitioned into $\lceil 3.5t+2\rceil$ such sets.
- Characterization of cycle obstruction sets for improper coloring planar graphs (with Ilkyoo Choi최일규 and Chun-Hung Liu), SIAM J. Discrete Math., **32(**2018), no. 2, pp. 1209-1228, 2018.For nonnegative integers $k, d_1, \ldots, d_k$, a graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ parts so that the $i$th part induces a graph with maximum degree at most $d_i$ for all $i\in\{1, \ldots, k\}$. A class $\mathcal C$ of graphs is {\it balanced $k$-partitionable} and {\it unbalanced $k$-partitionable} if there exists a nonnegative integer $D$ such that all graphs in $\mathcal C$ are $(D, \ldots, D)$-colorable and $(0, \ldots, 0, D)$-colorable, respectively, where the tuple has length $k$. A set $X$ of cycles is a {\it cycle obstruction set} of a class $\mathcal C$ of planar graphs if every planar graph containing none of the cycles in $X$ as a subgraph belongs to $\mathcal C$. This paper characterizes all cycle obstruction sets of planar graphs to be balanced $k$-partitionable and unbalanced $k$-partitionable for all $k$; namely, we identify all inclusion-wise minimal cycle obstruction sets for all $k$.
Characterization of cycle obstruction sets for improper coloring planar graphs
For nonnegative integers $k, d_1, \ldots, d_k$, a graph is $(d_1, \ldots, d_k)$-colorable if its vertex set can be partitioned into $k$ parts so that the $i$th part induces a graph with maximum degree at most $d_i$ for all $i\in\{1, \ldots, k\}$. A class $\mathcal C$ of graphs is {\it balanced $k$-partitionable} and {\it unbalanced $k$-partitionable} if there exists a nonnegative integer $D$ such that all graphs in $\mathcal C$ are $(D, \ldots, D)$-colorable and $(0, \ldots, 0, D)$-colorable, respectively, where the tuple has length $k$. A set $X$ of cycles is a {\it cycle obstruction set} of a class $\mathcal C$ of planar graphs if every planar graph containing none of the cycles in $X$ as a subgraph belongs to $\mathcal C$. This paper characterizes all cycle obstruction sets of planar graphs to be balanced $k$-partitionable and unbalanced $k$-partitionable for all $k$; namely, we identify all inclusion-wise minimal cycle obstruction sets for all $k$.
- Vertex-minors and the Erdős-Hajnal conjecture (with Maria Chudnovsky), Discrete Math., 341, pp. 3498-3499, 2018.We prove that for every graph $H$, there exists $\varepsilon>0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $|A|, |B|\ge \varepsilon n$ and $A$ is complete or anticomplete to $B$. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erdős-Hajnal conjecture for vertex-minors.
Vertex-minors and the Erdős-Hajnal conjecture
We prove that for every graph $H$, there exists $\varepsilon>0$ such that every $n$-vertex graph with no vertex-minors isomorphic to $H$ has a pair of disjoint sets $A$, $B$ of vertices such that $|A|, |B|\ge \varepsilon n$ and $A$ is complete or anticomplete to $B$. We deduce this from recent work of Chudnovsky, Scott, Seymour, and Spirkl (2018). This proves the analog of the Erdős-Hajnal conjecture for vertex-minors.
2017
- Rank-width: algorithmic and structural results, Discrete Applied Math., 231, pp. 15-24, November 2017.Rank-width is a width parameter of graphs describing whether it is possible to decompose a graph into a tree-like structure by `simple' cuts. This survey aims to summarize known algorithmic and structural results on rank-width of graphs.
Rank-width: algorithmic and structural results
Rank-width is a width parameter of graphs describing whether it is possible to decompose a graph into a tree-like structure by `simple' cuts. This survey aims to summarize known algorithmic and structural results on rank-width of graphs.
- The “art of trellis decoding” is fixed-parameter tractable (with Jisu Jeong정지수 and Eun Jung Kim김은정), IEEE Trans. Inform. Theory, 63, no. 11, pp. 7178-7205. (Previous title: constructive algorithm for path-width of matroids), November 2017.Given n subspaces of a finite-dimensional vector space over a fixed finite field $\mathbb F$, we wish to find a linear layout $V_1,V_2,\ldots,V_n$ of the subspaces such that $\dim((V_1+V_2+\cdots+V_i) \cap (V_{i+1}+\cdots+V_n))\le k$ for all i, such a linear layout is said to have width at most k. When restricted to 1-dimensional subspaces, this problem is equivalent to computing the trellis-width (or minimum trellis state-complexity) of a linear code in coding theory and computing the path-width of an $\mathbb F$-represented matroid in matroid theory. We present a fixed-parameter tractable algorithm to construct a linear layout of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over $\mathbb F$. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most k for an input $\mathbb F$-represented matroid of path-width at most k, and a fixed-parameter tractable algorithm to find a linear rank-decomposition of width at most k for an input graph of linear rank-width at most k. In both corollaries, no such algorithms were known previously. It was previously known that a fixed-parameter tractable algorithm exists for the decision version of the problem for matroid path-width, a theorem by Geelen, Gerards, and Whittle~(2002) implies that for each fixed finite field $\mathbb F$, there are finitely many forbidden $\mathbb F$-representable minors for the class of matroids of path-width at most k. An algorithm by Hliněný (2006) can detect a minor in an input $\mathbb F$-represented matroid of bounded branch-width. However, this indirect approach would not produce an actual path-decomposition. Our algorithm is the first one to construct such a path-decomposition and does not depend on the finiteness of forbidden minors.
The “art of trellis decoding” is fixed-parameter tractable
Given n subspaces of a finite-dimensional vector space over a fixed finite field $\mathbb F$, we wish to find a linear layout $V_1,V_2,\ldots,V_n$ of the subspaces such that $\dim((V_1+V_2+\cdots+V_i) \cap (V_{i+1}+\cdots+V_n))\le k$ for all i, such a linear layout is said to have width at most k. When restricted to 1-dimensional subspaces, this problem is equivalent to computing the trellis-width (or minimum trellis state-complexity) of a linear code in coding theory and computing the path-width of an $\mathbb F$-represented matroid in matroid theory. We present a fixed-parameter tractable algorithm to construct a linear layout of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over $\mathbb F$. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most k for an input $\mathbb F$-represented matroid of path-width at most k, and a fixed-parameter tractable algorithm to find a linear rank-decomposition of width at most k for an input graph of linear rank-width at most k. In both corollaries, no such algorithms were known previously. It was previously known that a fixed-parameter tractable algorithm exists for the decision version of the problem for matroid path-width, a theorem by Geelen, Gerards, and Whittle~(2002) implies that for each fixed finite field $\mathbb F$, there are finitely many forbidden $\mathbb F$-representable minors for the class of matroids of path-width at most k. An algorithm by Hliněný (2006) can detect a minor in an input $\mathbb F$-represented matroid of bounded branch-width. However, this indirect approach would not produce an actual path-decomposition. Our algorithm is the first one to construct such a path-decomposition and does not depend on the finiteness of forbidden minors.
- Even-cycle decompositions of graphs with no odd-K4-minor (with Tony Huynh and Maryam Verdian-Rizi), European J. Combin., 65, pp. 1-14, October 2017.An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no $K_5$-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-$K_4$-minor-free' cannot be replaced with `odd-$K_5$-minor-free.' The main technical ingredient is a structural characterization of the class of odd-$K_4$-minor-free graphs, which is due to Lovász, Seymour, Schrijver, and Truemper.
Even-cycle decompositions of graphs with no odd-K4-minor
An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no $K_5$-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-$K_4$-minor-free' cannot be replaced with `odd-$K_5$-minor-free.' The main technical ingredient is a structural characterization of the class of odd-$K_4$-minor-free graphs, which is due to Lovász, Seymour, Schrijver, and Truemper.
- Majority colouring of digraphs (with Stephan Kreutzer, Paul Seymour, David R. Wood, and Dominic van der Zypen), Electron. J. Combin., 24, #P2.25, May 2017.We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by 3.
Majority colouring of digraphs
We prove that every digraph has a vertex 4-colouring such that for each vertex $v$, at most half the out-neighbours of $v$ receive the same colour as $v$. We then obtain several results related to the conjecture obtained by replacing 4 by 3.
- Classification of real Bott manifolds and acyclic digraphs (with Suyoung Choi최수영 and Mikiya Masuda), Trans. Amer. Math. Soc., 369, no. 4, pp. 2987-3011, April 2017.We completely characterize real Bott manifolds up to affine diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $\mathbb Z/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. Our characterization can also be described in terms of graph operations on directed acyclic graphs. Using this combinatorial interpretation, we prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Finally, we produce some numerical invariants of real Bott manifolds from the viewpoint of graph theory and discuss their topological meaning. As a by-product, we prove that the toral rank conjecture holds for real Bott manifolds.
Classification of real Bott manifolds and acyclic digraphs
We completely characterize real Bott manifolds up to affine diffeomorphism in terms of three simple matrix operations on square binary matrices obtained from strictly upper triangular matrices by permuting rows and columns simultaneously. We also prove that any graded ring isomorphism between the cohomology rings of real Bott manifolds with $\mathbb Z/2$ coefficients is induced by an affine diffeomorphism between the real Bott manifolds. Our characterization can also be described in terms of graph operations on directed acyclic graphs. Using this combinatorial interpretation, we prove that the decomposition of a real Bott manifold into a product of indecomposable real Bott manifolds is unique up to permutations of the indecomposable factors. Finally, we produce some numerical invariants of real Bott manifolds from the viewpoint of graph theory and discuss their topological meaning. As a by-product, we prove that the toral rank conjecture holds for real Bott manifolds.
- Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors (with Ilkyoo Choi최일규 and O-joung Kwon권오정), J. Combin. Theory Ser. B, 123, pp. 126-147, March 2017.A fan $F_k$ is a graph that consists of an induced path on $k$ vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers $q$ and $k$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a vertex-minor isomorphic to $F_k$. We also prove that for all positive integers $q$ and $k\ge 3$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a pivot-minor isomorphic to a cycle of length $k$.
Coloring graphs without fan vertex-minors and graphs without cycle pivot-minors
A fan $F_k$ is a graph that consists of an induced path on $k$ vertices and an additional vertex that is adjacent to all vertices of the path. We prove that for all positive integers $q$ and $k$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a vertex-minor isomorphic to $F_k$. We also prove that for all positive integers $q$ and $k\ge 3$, every graph with sufficiently large chromatic number contains either a clique of size $q$ or a pivot-minor isomorphic to a cycle of length $k$.
- Strongly even-cycle decomposable graphs (with Tony Huynh, Andrew D. King, and Maryam Verdian-Rizi), J. Graph Theory, 84, no. 2, pp. 158-175, February 2017.A graph is strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs.
Strongly even-cycle decomposable graphs
A graph is strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs.
2016
- Dynamic coloring of graphs having no K5 minor (with Younjin Kim김연진 and Sang June Lee이상준), Discrete Applied Math., 206, pp. 81-89, June 2016.We prove that every simple connected graph with no $K_5$ minor admits a proper 4-coloring such that the neighborhood of each vertex $v$ having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result by S.-J. Kim, S. J. Lee, and W.-J. Park on planar graphs.
Dynamic coloring of graphs having no K5 minor
We prove that every simple connected graph with no $K_5$ minor admits a proper 4-coloring such that the neighborhood of each vertex $v$ having more than one neighbor is not monochromatic, unless the graph is isomorphic to the cycle of length 5. This generalizes the result by S.-J. Kim, S. J. Lee, and W.-J. Park on planar graphs.
- Unavoidable induced subgraphs in large graphs with no homogeneous sets (with Maria Chudnovsky, Ringi Kim김린기, and Paul Seymour), J. Combin. Theory Ser. B, 118, pp. 1-12, May 2016.A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$ is a sequence of $t+1$ vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors. We prove that for all $n$, there exists $N$ such that every prime graph with at least $N$ vertices contains one of the following graphs or their complements as an induced subgraph: (1) the graph obtained from $K_{1,n}$ by subdividing every edge once, (2) the line graph of $K_{2,n}$, (3) the line graph of the graph in (1), (4) the half-graph of height $n$, (5) a prime graph induced by a chain of length $n$, (6) two particular graphs obtained from the half-graph of height $n$ by making one side a clique and adding one vertex.
Unavoidable induced subgraphs in large graphs with no homogeneous sets
A homogeneous set of an $n$-vertex graph is a set $X$ of vertices ($2\le |X|\le n-1$) such that every vertex not in $X$ is either complete or anticomplete to $X$. A graph is called prime if it has no homogeneous set. A chain of length $t$ is a sequence of $t+1$ vertices such that for every vertex in the sequence except the first one, its immediate predecessor is its unique neighbor or its unique non-neighbor among all of its predecessors. We prove that for all $n$, there exists $N$ such that every prime graph with at least $N$ vertices contains one of the following graphs or their complements as an induced subgraph: (1) the graph obtained from $K_{1,n}$ by subdividing every edge once, (2) the line graph of $K_{2,n}$, (3) the line graph of the graph in (1), (4) the half-graph of height $n$, (5) a prime graph induced by a chain of length $n$, (6) two particular graphs obtained from the half-graph of height $n$ by making one side a clique and adding one vertex.
2015
- Number of cliques in graphs with a forbidden subdivision (with Choongbum Lee이중범), SIAM J. Discrete Math., 29, no. 4, pp. 1999-2005, October 2015.We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to zero as $t$ tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in $n$-vertex graphs with no $K_t$-minor is at most $2^{ct}n$ for some constant $c$.
Number of cliques in graphs with a forbidden subdivision
We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to zero as $t$ tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in $n$-vertex graphs with no $K_t$-minor is at most $2^{ct}n$ for some constant $c$.
- A relative of Hadwiger’s conjecture (with Katherine Edwards, Dong Yeap Kang강동엽, Jaehoon Kim김재훈, and Paul Seymour), SIAM J. Discrete Math., 29, no. 4, pp. 2385–2388, 2015.Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots, X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property.
A relative of Hadwiger’s conjecture
Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned into $t$ sets $X_1,\ldots, X_t$, such that for $1\le i\le t$, the subgraph induced on $X_i$ has maximum degree at most a function of $t$. This is sharp, in that the conclusion becomes false if we ask for a partition into $t-1$ sets with the same property.
2014
- Hyperbolic surface subgroups of one-ended doubles of free groups (with Sang-hyun Kim김상현), J. Topology, 7, no. 4, pp. 927-947, December 2014.Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank two, or (2) every generator is used the same number of times in the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).
Hyperbolic surface subgroups of one-ended doubles of free groups
Gromov asked whether every one-ended word-hyperbolic group contains a hyperbolic surface group. We prove that every one-ended double of a free group has a hyperbolic surface subgroup if (1) the free group has rank two, or (2) every generator is used the same number of times in the amalgamating words. To prove this, we formulate a stronger statement on Whitehead graphs and prove its specialization by combinatorial induction for (1) and the characterization of perfect matching polytopes by Edmonds for (2).
- Excluded vertex-minors for graphs of linear rank-width at most k (with Jisu Jeong정지수 and O-joung Kwon권오정), European J. Combin., 41, pp. 242-257, October 2014.Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each $k$, there is a finite obstruction set $\mathcal{O}_k$ of graphs such that a graph $G$ has linear rank-width at most $k$ if and only if no vertex-minor of $G$ is isomorphic to a graph in $\mathcal{O}_k$. However, no attempts have been made to bound the number of graphs in $\mathcal{O}_k$ for $k\ge 2$. We show that for each $k$, there are at least $2^{Ω(3^k)}$ pairwise locally non-equivalent graphs in $\mathcal{O}_k$, and therefore the number of graphs in $\mathcal{O}_k$ is at least double exponential. To prove this theorem, it is necessary to characterize when two graphs in $\mathcal O_k$ are locally equivalent. A graph is a block graph if all of its blocks are complete graphs. We prove that if two block graphs without simplicial vertices of degree at least $2$ are locally equivalent, then they are isomorphic. This not only is useful for our theorem but also implies a theorem of Bouchet [Transforming trees by successive local complementations, J. Graph Theory 12 (1988), no. 2, 195-207] stating that if two trees are locally equivalent, then they are isomorphic.
Excluded vertex-minors for graphs of linear rank-width at most k
Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each $k$, there is a finite obstruction set $\mathcal{O}_k$ of graphs such that a graph $G$ has linear rank-width at most $k$ if and only if no vertex-minor of $G$ is isomorphic to a graph in $\mathcal{O}_k$. However, no attempts have been made to bound the number of graphs in $\mathcal{O}_k$ for $k\ge 2$. We show that for each $k$, there are at least $2^{Ω(3^k)}$ pairwise locally non-equivalent graphs in $\mathcal{O}_k$, and therefore the number of graphs in $\mathcal{O}_k$ is at least double exponential. To prove this theorem, it is necessary to characterize when two graphs in $\mathcal O_k$ are locally equivalent. A graph is a block graph if all of its blocks are complete graphs. We prove that if two block graphs without simplicial vertices of degree at least $2$ are locally equivalent, then they are isomorphic. This not only is useful for our theorem but also implies a theorem of Bouchet [Transforming trees by successive local complementations, J. Graph Theory 12 (1988), no. 2, 195-207] stating that if two trees are locally equivalent, then they are isomorphic.
- Unavoidable vertex-minors in large prime graphs (with O-joung Kwon권오정), European J. Combin., 41, pp. 100-127, October 2014.A graph is prime (with respect to the split decomposition) if its vertex set does not admit a partition (A,B) (called a split) with |A|, |B| >= 2 such that the set of edges joining A and B induces a complete bipartite graph. We prove that for each n, there exists N such that every prime graph on at least N vertices contains a vertex-minor isomorphic to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching.
Unavoidable vertex-minors in large prime graphs
A graph is prime (with respect to the split decomposition) if its vertex set does not admit a partition (A,B) (called a split) with |A|, |B| >= 2 such that the set of edges joining A and B induces a complete bipartite graph. We prove that for each n, there exists N such that every prime graph on at least N vertices contains a vertex-minor isomorphic to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching.
- Faster algorithms for vertex partitioning problems parameterized by clique-width (with Sigve Hortemo Sæther and Martin Vatshelle), Theoret. Comput. Sci., 535, pp. 16-24, May 2014.Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width $k$ given with a $k$-expression, Dominating Set can be solved in $4^k n^{O(1)}$ time. However, no FPT algorithm is known for computing an optimal $k$-expression. For a graph of clique-width $k$, if we rely on known algorithms to compute a $(2^{3k}-1)$-expression via rank-width and then solving Dominating Set using the $(2^{3k}-1)$-expression, the above algorithm will only give a runtime of $4^{2^{3k}} n^{O(1)}$. There have been results which overcome this exponential jump; the best known algorithm can solve Dominating Set in time $2^{O(k^2)} n^{O(1)}$ by avoiding constructing a $k$-expression [Bui-Xuan, Telle, and Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoret. Comput. Sci., 2013. doi:10.1016/j.tcs.2013.01.009]. We improve this to $2^{O(k\log k)}n^{O(1)}$. Indeed, we show that for a graph of clique-width $k$, a large class of domination and partitioning problems (LC-VSP), including Dominating Set, can be solved in $2^{O(k\log{k})} n^{O(1)}$. Our main tool is a variant of rank-width using the rank of a $0$-$1$ matrix over the rational field instead of the binary field.
Faster algorithms for vertex partitioning problems parameterized by clique-width
Many NP-hard problems, such as Dominating Set, are FPT parameterized by clique-width. For graphs of clique-width $k$ given with a $k$-expression, Dominating Set can be solved in $4^k n^{O(1)}$ time. However, no FPT algorithm is known for computing an optimal $k$-expression. For a graph of clique-width $k$, if we rely on known algorithms to compute a $(2^{3k}-1)$-expression via rank-width and then solving Dominating Set using the $(2^{3k}-1)$-expression, the above algorithm will only give a runtime of $4^{2^{3k}} n^{O(1)}$. There have been results which overcome this exponential jump; the best known algorithm can solve Dominating Set in time $2^{O(k^2)} n^{O(1)}$ by avoiding constructing a $k$-expression [Bui-Xuan, Telle, and Vatshelle. Fast dynamic programming for locally checkable vertex subset and vertex partitioning problems. Theoret. Comput. Sci., 2013. doi:10.1016/j.tcs.2013.01.009]. We improve this to $2^{O(k\log k)}n^{O(1)}$. Indeed, we show that for a graph of clique-width $k$, a large class of domination and partitioning problems (LC-VSP), including Dominating Set, can be solved in $2^{O(k\log{k})} n^{O(1)}$. Our main tool is a variant of rank-width using the rank of a $0$-$1$ matrix over the rational field instead of the binary field.
- Graphs of small rank-width are pivot-minors of graphs of small tree-width (with O-joung Kwon권오정), Discrete Applied Math., 168, pp. 108-118, May 2014.We prove that every graph of rank-width $k$ is a pivot-minor of a graph of tree-width at most $2k$. We also prove that graphs of rank-width at most 1, equivalently distance-hereditary graphs, are exactly vertex-minors of trees, and graphs of linear rank-width at most 1 are precisely vertex-minors of paths. In addition, we show that bipartite graphs of rank-width at most 1 are exactly pivot-minors of trees and bipartite graphs of linear rank-width at most 1 are precisely pivot-minors of paths.
Graphs of small rank-width are pivot-minors of graphs of small tree-width
We prove that every graph of rank-width $k$ is a pivot-minor of a graph of tree-width at most $2k$. We also prove that graphs of rank-width at most 1, equivalently distance-hereditary graphs, are exactly vertex-minors of trees, and graphs of linear rank-width at most 1 are precisely vertex-minors of paths. In addition, we show that bipartite graphs of rank-width at most 1 are exactly pivot-minors of trees and bipartite graphs of linear rank-width at most 1 are precisely pivot-minors of paths.
2012
- Rank-width of Random Graphs (with Choongbum Lee이중범 and Joonkyung Lee이준경), J. Graph Theory, 70, no. 3, pp. 339-347, July 2012.Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i) if 0<p<1 is a constant, then rw(G(n,p)) = \lceil n/3 \rceil-O(1), (ii) if 1/n<< p <1/2, then rw(G(n,p))= \lceil n/3\rceil-o(n), (iii) if p = c/n and c > 1, then rw(G(n,p)) > r n for some r = r(c), and (iv) if p <= c/n and c<1, then rw(G(n,p)) <=2. As a corollary, we deduce that G(n,p) has linear tree-width whenever p=c/n for each c>1, answering a question of Gao (2006).
Rank-width of Random Graphs
Rank-width of a graph G, denoted by rw(G), is a width parameter of graphs introduced by Oum and Seymour (2006). We investigate the asymptotic behavior of rank-width of a random graph G(n,p). We show that, asymptotically almost surely, (i) if 0<p<1 is a constant, then rw(G(n,p)) = \lceil n/3 \rceil-O(1), (ii) if 1/n<< p <1/2, then rw(G(n,p))= \lceil n/3\rceil-o(n), (iii) if p = c/n and c > 1, then rw(G(n,p)) > r n for some r = r(c), and (iv) if p <= c/n and c<1, then rw(G(n,p)) <=2. As a corollary, we deduce that G(n,p) has linear tree-width whenever p=c/n for each c>1, answering a question of Gao (2006).
- Finding minimum clique capacity (with Maria Chudnovsky and Paul Seymour), Combinatorica, 32, no. 3, pp. 283-287, April 2012.Let C be a clique of a graph G. The capacity of C is defined to be (|V (G) \ C| + |D|)/2, where D is the set of vertices in V (G) \ C that have both a neighbour and a non-neighbour in C. We give a polynomial-time algorithm to find the minimum clique capacity in a graph G. This problem arose as an open question in a study [1] of packing vertex-disjoint induced three-vertex paths in a graph with no stable set of size three, which in turn was motivated by Hadwiger’s conjecture.
Finding minimum clique capacity
Let C be a clique of a graph G. The capacity of C is defined to be (|V (G) \ C| + |D|)/2, where D is the set of vertices in V (G) \ C that have both a neighbour and a non-neighbour in C. We give a polynomial-time algorithm to find the minimum clique capacity in a graph G. This problem arose as an open question in a study [1] of packing vertex-disjoint induced three-vertex paths in a graph with no stable set of size three, which in turn was motivated by Hadwiger’s conjecture.
- Rank-width and Well-quasi-ordering of skew-symmetric or symmetric matrices, Linear Algebra Appl., 436(April 1, 2012), no. 7, pp. 2008-2036, April 2012.We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2,... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_j, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour's theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum's theorem for graphs of bounded rank-width with respect to pivot-minors.
Rank-width and Well-quasi-ordering of skew-symmetric or symmetric matrices
We prove that every infinite sequence of skew-symmetric or symmetric matrices M_1, M_2,... over a fixed finite field must have a pair M_i, M_j (i<j) such that M_i is isomorphic to a principal submatrix of the Schur complement of a nonsingular principal submatrix in M_j, if those matrices have bounded rank-width. This generalizes three theorems on well-quasi-ordering of graphs or matroids admitting good tree-like decompositions; (1) Robertson and Seymour's theorem for graphs of bounded tree-width, (2) Geelen, Gerards, and Whittle's theorem for matroids representable over a fixed finite field having bounded branch-width, and (3) Oum's theorem for graphs of bounded rank-width with respect to pivot-minors.
2011
- Perfect Matchings in Claw-free Cubic Graphs, Electron. J. Combin., 18, #P62 (pp. 6), 2011.Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.
Perfect Matchings in Claw-free Cubic Graphs
Lovasz and Plummer conjectured that there exists a fixed positive constant c such that every cubic n-vertex graph with no cutedge has at least 2^(cn) perfect matchings. Their conjecture has been verified for bipartite graphs by Voorhoeve and planar graphs by Chudnovsky and Seymour. We prove that every claw-free cubic n-vertex graph with no cutedge has more than 2^(n/12) perfect matchings, thus verifying the conjecture for claw-free graphs.
2010
- Rank-width and tree-width of H-minor-free graphs (with Fedor V. Fomin and Dimitrios M. Thilikos), European J. Combin., 31, no. 7, pp. 1617-1628, 2010.We prove that for any fixed r>=2, the tree-width of graphs not containing K_r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as K_r-minor free graphs and graphs of bounded genus.
Rank-width and tree-width of H-minor-free graphs
We prove that for any fixed r>=2, the tree-width of graphs not containing K_r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as K_r-minor free graphs and graphs of bounded genus.
2009
- Circle Graph Obstructions under Pivoting (with Jim Geelen), J. Graph Theory, 61, no. 1, pp. 1-11, 2009.A circle graph is the intersection graph of a set of chords of a circle. The class of circle graphs is closed under pivot‐minors. We determine the pivot‐minor‐minimal non‐circle‐graphs; there are 15 obstructions. These obstructions are found, by computer search, as a corollary to Bouchet's characterization of circle graphs under local complementation. Our characterization generalizes Kuratowski's Theorem.
Circle Graph Obstructions under Pivoting
A circle graph is the intersection graph of a set of chords of a circle. The class of circle graphs is closed under pivot‐minors. We determine the pivot‐minor‐minimal non‐circle‐graphs; there are 15 obstructions. These obstructions are found, by computer search, as a corollary to Bouchet's characterization of circle graphs under local complementation. Our characterization generalizes Kuratowski's Theorem.
- Computing rank-width exactly, Information Proc. Letters, 109, no. 13, pp. 745-748, 2009.We prove that the rank-width of an n-vertex graph can be com- puted exactly in time O(2n n3 log2 n log log n). To improve over a trivial O(3n log n)-time algorithm, we develop a general framework for decom- positions on which an optimal decomposition can be computed efficiently. This framework may be used for other width parameters, including the branch-width of matroids and the carving-width of graphs.
Computing rank-width exactly
We prove that the rank-width of an n-vertex graph can be com- puted exactly in time O(2n n3 log2 n log log n). To improve over a trivial O(3n log n)-time algorithm, we develop a general framework for decom- positions on which an optimal decomposition can be computed efficiently. This framework may be used for other width parameters, including the branch-width of matroids and the carving-width of graphs.
- Excluding a Bipartite Circle Graph from Line Graphs, J. Graph Theory, 60, no. 3, pp. 183-203, 2009.We prove that, for a fixed bipartite circle graph H, all line graphs with sufficiently large rank‐width (or clique‐width) must have a pivot‐minor isomorphic to H. To prove this, we introduce graphic delta‐matroids. Graphic delta‐matroids are minors of delta‐matroids of line graphs and they generalize graphic and cographic matroids.
Excluding a Bipartite Circle Graph from Line Graphs
We prove that, for a fixed bipartite circle graph H, all line graphs with sufficiently large rank‐width (or clique‐width) must have a pivot‐minor isomorphic to H. To prove this, we introduce graphic delta‐matroids. Graphic delta‐matroids are minors of delta‐matroids of line graphs and they generalize graphic and cographic matroids.
2008
- Approximating rank-width and clique-width quickly, ACM Trans. Algorithms, 5. no. 1, Art. No. 10, 2008.Rank-width was defined by Oum and Seymour [2006] to investigate clique-width. They constructed an algorithm that either outputs a rank-decomposition of width at most f ( k ) for some function f or confirms that rank-width is larger than k in time O (| V | 9 log | V |) for an input graph G = ( V, E ) and a fixed k. We develop three separate algorithms of this kind with faster running time. We construct an O (| V | 4 )-time algorithm with f ( k ) = 3 k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O (| V | 3 )-time algorithm with f ( k ) = 24 k, achieved by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný [2005]. Finally we construct an O (| V | 3 )-time algorithm with f ( k ) = 3 k − 1 by combining the ideas of the two previously cited papers.
Approximating rank-width and clique-width quickly
Rank-width was defined by Oum and Seymour [2006] to investigate clique-width. They constructed an algorithm that either outputs a rank-decomposition of width at most f ( k ) for some function f or confirms that rank-width is larger than k in time O (| V | 9 log | V |) for an input graph G = ( V, E ) and a fixed k. We develop three separate algorithms of this kind with faster running time. We construct an O (| V | 4 )-time algorithm with f ( k ) = 3 k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O (| V | 3 )-time algorithm with f ( k ) = 24 k, achieved by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný [2005]. Finally we construct an O (| V | 3 )-time algorithm with f ( k ) = 3 k − 1 by combining the ideas of the two previously cited papers.
- Finding Branch-decompositions and Rank-decompositions (with Petr Hliněný), SIAM J. Comput., 38, no. 3, pp. 1012-1032, 2008.We present a new algorithm that can output the rank-decomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixed-parameter tractable, that is, they run in time O(n3 ) for each fixed value of k where n is the number of vertices / elements of the input. (The previous best algorithm for construction of a branch-decomposition or a rank-decomposition of optimal width due to Oum and Seymour [Testing branch-width. J. Combin. Theory Ser. B, 97(3) (2007) 385–393] is not fixed-parameter tractable.)
Finding Branch-decompositions and Rank-decompositions
We present a new algorithm that can output the rank-decomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixed-parameter tractable, that is, they run in time O(n3 ) for each fixed value of k where n is the number of vertices / elements of the input. (The previous best algorithm for construction of a branch-decomposition or a rank-decomposition of optimal width due to Oum and Seymour [Testing branch-width. J. Combin. Theory Ser. B, 97(3) (2007) 385–393] is not fixed-parameter tractable.)
- Rank-width and Well-quasi-ordering, SIAM J. Discrete Math., 22, no. 2, pp. 666-682, 2008.Robertson and Seymour (1990) proved that graphs of bounded tree-width are well-quasi-ordered by the graph minor relation. By extending their arguments, Geelen, Gerards, and Whittle (2002) proved that binary matroids of bounded branch-width are well-quasi-ordered by the matroid minor relation. We prove another theorem of this kind in terms of rank-width and vertex-minors. For a graph G = (V, E) and a vertex v of G, a local complementation at v is an operation that replaces the subgraph induced by the neighbors of v with its complement graph. A graph H is called a vertex-minor of G if H can be obtained from G by applying a sequence of vertex deletions and local complementations. Rank-width was defined by Oum and Seymour (2006) to investigate clique-width; they showed that graphs have bounded rank-width if and only if they have bounded clique-width. We prove that graphs of bounded rank-width are well-quasi-ordered by the vertex-minor relation; in other words, for every infinite sequence G1, G2,... of graphs of rank-width (or clique-width) at most k, there exist i < j such that Gi is isomorphic to a vertex-minor of Gj. This implies that there is a finite list of graphs such that a graph has rank-width at most k if and only if it contains no one in the list as a vertex-minor. The proof uses the notion of isotropic systems defined by Bouchet.
Rank-width and Well-quasi-ordering
Robertson and Seymour (1990) proved that graphs of bounded tree-width are well-quasi-ordered by the graph minor relation. By extending their arguments, Geelen, Gerards, and Whittle (2002) proved that binary matroids of bounded branch-width are well-quasi-ordered by the matroid minor relation. We prove another theorem of this kind in terms of rank-width and vertex-minors. For a graph G = (V, E) and a vertex v of G, a local complementation at v is an operation that replaces the subgraph induced by the neighbors of v with its complement graph. A graph H is called a vertex-minor of G if H can be obtained from G by applying a sequence of vertex deletions and local complementations. Rank-width was defined by Oum and Seymour (2006) to investigate clique-width; they showed that graphs have bounded rank-width if and only if they have bounded clique-width. We prove that graphs of bounded rank-width are well-quasi-ordered by the vertex-minor relation; in other words, for every infinite sequence G1, G2,... of graphs of rank-width (or clique-width) at most k, there exist i < j such that Gi is isomorphic to a vertex-minor of Gj. This implies that there is a finite list of graphs such that a graph has rank-width at most k if and only if it contains no one in the list as a vertex-minor. The proof uses the notion of isotropic systems defined by Bouchet.
- Rank-width is less than or equal to branch-width, J. Graph Theory, 57, no. 3, pp. 239-244, 2008.We prove that the rank‐width of the incidence graph of a graph G is either equal to or exactly one less than the branch‐width of G, unless the maximum degree of G is 0 or 1. This implies that rank‐width of a graph is less than or equal to branch‐width of the graph unless the branch‐width is 0. Moreover, this inequality is tight.
Rank-width is less than or equal to branch-width
We prove that the rank‐width of the incidence graph of a graph G is either equal to or exactly one less than the branch‐width of G, unless the maximum degree of G is 0 or 1. This implies that rank‐width of a graph is less than or equal to branch‐width of the graph unless the branch‐width is 0. Moreover, this inequality is tight.
- Width Parameters Beyond Tree-width and Their Applications (with Petr Hliněný, Detlef Seese, and Georg Gottlob), The Computer Journal, 51, no. 3, pp. 326-362, 2008.Besides the successful concept of tree-width (see [H. Bodlaender, A. Koster: Combinatorial optimisation on graphs of bounded treewidth, ****** this survey volume ******, 14 p.]) in the past years, many con- cepts and parameters measuring a similarity of structures to trees, or how a structure distinguishes from a tree, have been born and studied. These concepts and parameters proved to be useful tools for many applications, especially in the design of efficient algorithms. We present a novel view of contemporary developments of these “width” parameters in combinatorial structures that, besides traditional tree-width and derived dynamic pro- gramming schemes, leads to other usable parameters like branch-width, ∗ Corresponding author, hlineny@fi.muni.cz. 1 rank-width (clique-width), or hypertree-width. Our article demonstrates how “width” parameters of graphs and generalized structures (like ma- troids or hypergraphs), on an abstract level, can be used to improve the design of parameterized algorithms and the structural analysis in other applications.
Width Parameters Beyond Tree-width and Their Applications
Besides the successful concept of tree-width (see [H. Bodlaender, A. Koster: Combinatorial optimisation on graphs of bounded treewidth, ****** this survey volume ******, 14 p.]) in the past years, many con- cepts and parameters measuring a similarity of structures to trees, or how a structure distinguishes from a tree, have been born and studied. These concepts and parameters proved to be useful tools for many applications, especially in the design of efficient algorithms. We present a novel view of contemporary developments of these “width” parameters in combinatorial structures that, besides traditional tree-width and derived dynamic pro- gramming schemes, leads to other usable parameters like branch-width, ∗ Corresponding author, hlineny@fi.muni.cz. 1 rank-width (clique-width), or hypertree-width. Our article demonstrates how “width” parameters of graphs and generalized structures (like ma- troids or hypergraphs), on an abstract level, can be used to improve the design of parameterized algorithms and the structural analysis in other applications.
2007
- Testing Branch-width (with Paul Seymour), J. Combin. Theory Ser. B, 97, no. 3, pp. 385-393., 2007. Corrigendum to our paper “Testing branch-width”.An integer-valued function f on the set 2V of all subsets of a finite set V is a connectivity function if it satisfies the following conditions: (1) f (X) + f (Y ) ≥ f (X ∩ Y ) + f (X ∪ Y ) for all subsets X, Y of V, (2) f (X) = f (V \ X) for all X ⊆ V, and (3) f (∅) = 0. Branch-width is defined for graphs, matroids, and more generally, connectivity functions. We show that for each constant k, there is a polynomial-time (in |V |) algorithm to decide whether the branch-width of a connectivity function f is at most k, if f is given by an oracle. This algorithm can be applied to branch-width, carving-width, and rank-width of graphs. In particular, we can recognize matroids M of branch-width at most k in polynomial (in |E(M)|) time if the matroid is given by an independence oracle.
Testing Branch-width
An integer-valued function f on the set 2V of all subsets of a finite set V is a connectivity function if it satisfies the following conditions: (1) f (X) + f (Y ) ≥ f (X ∩ Y ) + f (X ∪ Y ) for all subsets X, Y of V, (2) f (X) = f (V \ X) for all X ⊆ V, and (3) f (∅) = 0. Branch-width is defined for graphs, matroids, and more generally, connectivity functions. We show that for each constant k, there is a polynomial-time (in |V |) algorithm to decide whether the branch-width of a connectivity function f is at most k, if f is given by an oracle. This algorithm can be applied to branch-width, carving-width, and rank-width of graphs. In particular, we can recognize matroids M of branch-width at most k in polynomial (in |E(M)|) time if the matroid is given by an independence oracle.
- Vertex-Minors, Monadic Second-Order Logic, and a Conjecture by Seese (with Bruno Courcelle), J. Combin. Theory Ser. B, 97, no. 1, pp. 91-126, 2007.We prove that one can express the vertex-minor relation on finite undirected graphs by formulas of monadic second-order logic (with no edge set quantification) extended with a predicate expressing that a set has even cardinality. We obtain a slight weakening of a conjecture by Seese stating that sets of graphs having a decidable satisfiability problem for monadic second-order logic have bounded clique-width. We also obtain a polynomial-time algorithm to check that the rank-width of a graph is at most k for any fixed k. The proofs use isotropic systems.
Vertex-Minors, Monadic Second-Order Logic, and a Conjecture by Seese
We prove that one can express the vertex-minor relation on finite undirected graphs by formulas of monadic second-order logic (with no edge set quantification) extended with a predicate expressing that a set has even cardinality. We obtain a slight weakening of a conjecture by Seese stating that sets of graphs having a decidable satisfiability problem for monadic second-order logic have bounded clique-width. We also obtain a polynomial-time algorithm to check that the rank-width of a graph is at most k for any fixed k. The proofs use isotropic systems.
2006
- Approximating Clique-width and Branch-width (with Paul Seymour), J. Combin. Theory Ser. B, 96, no. 4, pp. 514-528., 2006. (the 9th in the most downloaded papers between Jan-Mar 2006 from JCT B) (In the list of classic papers in Discrete Mathematics of 2006 by Google Scholar).We construct a polynomial-time algorithm to approximate the branch-width of certain symmetric submodular functions, and give two applications. The first is to graph “clique-width”. Clique-width is a measure of the difficulty of decom- posing a graph in a kind of tree-structure, and if a graph has clique-width at most k then the corresponding decomposition of the graph is called a “k-expression”. We find (for fixed k) an O(n9 log n)-time algorithm that, with input an n-vertex graph, outputs either a (23k+2 − 1)- expression for the graph, or a true statement that the graph has clique-width at least k + 1. (The best earlier algorithm algorithm, by Johansson [13], constructed a 2k log n-expression for graphs of clique-width at most k.) It was already known that several graph problems, NP- hard on general graphs, are solvable in polynomial time if the input graph comes equipped with a k-expression (for fixed k). As a consequence of our algorithm, the same conclusion follows under the weaker hypothesis that the input graph has clique-width at most k (thus, we no longer need to be provided with an explicit k-expression). Another application is to the area of matroid branch-width. For fixed k, we find an O(n4 )- time algorithm that, with input an n-element matroid in terms of its rank oracle, either outputs a branch-decomposition of width at most 3k − 1 or a true statement that the ma- troid has branch-width at least k + 1. The previous algorithm by Hliněný [11] was only for representable matroids.
Approximating Clique-width and Branch-width
We construct a polynomial-time algorithm to approximate the branch-width of certain symmetric submodular functions, and give two applications. The first is to graph “clique-width”. Clique-width is a measure of the difficulty of decom- posing a graph in a kind of tree-structure, and if a graph has clique-width at most k then the corresponding decomposition of the graph is called a “k-expression”. We find (for fixed k) an O(n9 log n)-time algorithm that, with input an n-vertex graph, outputs either a (23k+2 − 1)- expression for the graph, or a true statement that the graph has clique-width at least k + 1. (The best earlier algorithm algorithm, by Johansson [13], constructed a 2k log n-expression for graphs of clique-width at most k.) It was already known that several graph problems, NP- hard on general graphs, are solvable in polynomial time if the input graph comes equipped with a k-expression (for fixed k). As a consequence of our algorithm, the same conclusion follows under the weaker hypothesis that the input graph has clique-width at most k (thus, we no longer need to be provided with an explicit k-expression). Another application is to the area of matroid branch-width. For fixed k, we find an O(n4 )- time algorithm that, with input an n-element matroid in terms of its rank oracle, either outputs a branch-decomposition of width at most 3k − 1 or a true statement that the ma- troid has branch-width at least k + 1. The previous algorithm by Hliněný [11] was only for representable matroids.
2005
- Rank-width and Vertex-minors, J. Combin. Theory Ser. B, 95, no. 1, pp. 79-100, 2005. Corrigendum to: “Rank-width and vertex-minors”, 2009. (the 9th in the most downloaded papers between Jul-Sep 2005 from JCT B).The rank-width is a graph parameter related in terms of fixed functions to clique- width but more tractable. Clique-width has nice algorithmic properties, but no good “minor” relation is known analogous to graph minor embedding for tree-width. In this paper, we discuss the vertex-minor relation of graphs and its connection with rank-width. We prove a relationship between vertex-minors of bipartite graphs and minors of binary matroids, and as an application, we prove that bipartite graphs of sufficiently large rank-width contain certain bipartite graphs as vertex-minors. The main theorem of this paper is that for fixed k, there is a finite list of graphs such that a graph G has rank-width at most k if and only if no graph in the list is isomorphic to a vertex-minor of G. Furthermore, we prove that a graph has rank-width at most 1 if and only if it is distance-hereditary.
Rank-width and Vertex-minors
The rank-width is a graph parameter related in terms of fixed functions to clique- width but more tractable. Clique-width has nice algorithmic properties, but no good “minor” relation is known analogous to graph minor embedding for tree-width. In this paper, we discuss the vertex-minor relation of graphs and its connection with rank-width. We prove a relationship between vertex-minors of bipartite graphs and minors of binary matroids, and as an application, we prove that bipartite graphs of sufficiently large rank-width contain certain bipartite graphs as vertex-minors. The main theorem of this paper is that for fixed k, there is a finite list of graphs such that a graph G has rank-width at most k if and only if no graph in the list is isomorphic to a vertex-minor of G. Furthermore, we prove that a graph has rank-width at most 1 if and only if it is distance-hereditary.
Ph.D. Thesis
- We define rank-width of graphs to investigate clique-width. Rank-width is a complexity measure of decomposing a graph in a kind of tree-structure, called a rank-decomposition. We show that graphs have bounded rank-width if and only if they have bounded clique-width. It is unknown how to recognize graphs of clique-width at most k for fixed k > 3 in polynomial time. However, we find an algorithm recognizing graphs of rank-width at most k, by combining following three ingredients. First, we construct a polynomial-time algorithm, for fixed k, that confirms rank-width is larger than k or outputs a rank-decomposition of width at most f (k) for some function f. It was known that many hard graph problems have polynomial-time algorithms for graphs of bounded clique-width, however, requiring a given decomposition corresponding to clique-width (k-expression ); we remove this requirement. Second, we define graph vertex-minors which generalizes matroid minors, and prove that if {G1, G2,…} is an infinite sequence of graphs of bounded rank-width, then there exist i < j such that Gi is isomorphic to a vertex-minor of Gj. Consequently there is a finite list Ck of graphs such that a graph has rank-width at most k if and only if none of its vertex-minors are isomorphic to a graph in Ck. Finally we construct, for fixed graph H, a modulo-2 counting monadic second-order logic formula expressing a graph contains a vertex-minor isomorphic to H. It is known that such logic formulas are solvable in linear time on graphs of bounded clique-width if the k-expression is given as an input. Another open problem in the area of clique-width is Seese's conjecture; if a set of graphs have an algorithm to answer whether a given monadic second-order logic formula is true for all graphs in the set, then it has bounded rank-width. We prove a weaker statement; if the algorithm answers for all modulo-2 counting monadic second-order logic formulas, then the set has bounded rank-width.
Graphs of Bounded Rank-width
We define rank-width of graphs to investigate clique-width. Rank-width is a complexity measure of decomposing a graph in a kind of tree-structure, called a rank-decomposition. We show that graphs have bounded rank-width if and only if they have bounded clique-width. It is unknown how to recognize graphs of clique-width at most k for fixed k > 3 in polynomial time. However, we find an algorithm recognizing graphs of rank-width at most k, by combining following three ingredients. First, we construct a polynomial-time algorithm, for fixed k, that confirms rank-width is larger than k or outputs a rank-decomposition of width at most f (k) for some function f. It was known that many hard graph problems have polynomial-time algorithms for graphs of bounded clique-width, however, requiring a given decomposition corresponding to clique-width (k-expression ); we remove this requirement. Second, we define graph vertex-minors which generalizes matroid minors, and prove that if {G1, G2,…} is an infinite sequence of graphs of bounded rank-width, then there exist i < j such that Gi is isomorphic to a vertex-minor of Gj. Consequently there is a finite list Ck of graphs such that a graph has rank-width at most k if and only if none of its vertex-minors are isomorphic to a graph in Ck. Finally we construct, for fixed graph H, a modulo-2 counting monadic second-order logic formula expressing a graph contains a vertex-minor isomorphic to H. It is known that such logic formulas are solvable in linear time on graphs of bounded clique-width if the k-expression is given as an input. Another open problem in the area of clique-width is Seese's conjecture; if a set of graphs have an algorithm to answer whether a given monadic second-order logic formula is true for all graphs in the set, then it has bounded rank-width. We prove a weaker statement; if the algorithm answers for all modulo-2 counting monadic second-order logic formulas, then the set has bounded rank-width.
Conference Papers
2026
- Branch-width of connectivity functions is fixed-parameter tractable (with Tuukka Korhonen), In the Proceedings of the 67th Annual Symposium on Foundations of Computer Science (FOCS 2026, New York City, NY, USA, November 8-11, 2026), accepted, November 2026.A connectivity function on a finite set $V$ is a symmetric submodular function $f \colon 2^V \to \mathbb{Z}$ with $f(\emptyset)=0$. We prove that finding a branch-decomposition of width at most $k$ for a connectivity function given by an oracle is fixed-parameter tractable (FPT), by providing an algorithm of running time $2^{O(k^2)} γn^6 \log n$, where $γ$ is the time to compute $f(X)$ for any set $X$, and $n = |V|$. This improves the previous algorithm by Oum and Seymour [J. Combin. Theory Ser. B, 2007], which runs in time $γn^{O(k)}$. Our algorithm can be applied to rank-width of graphs, branch-width of matroids, branch-width of (hyper)graphs, and carving-width of graphs. This resolves an open problem asked by Hliněný [SIAM J. Comput., 2005], who asked whether branch-width of matroids given by the rank oracle is fixed-parameter tractable. Furthermore, our algorithm improves the best known dependency on $k$ in the running times of FPT algorithms for graph branch-width, rank-width, and carving-width.
Branch-width of connectivity functions is fixed-parameter tractable
A connectivity function on a finite set $V$ is a symmetric submodular function $f \colon 2^V \to \mathbb{Z}$ with $f(\emptyset)=0$. We prove that finding a branch-decomposition of width at most $k$ for a connectivity function given by an oracle is fixed-parameter tractable (FPT), by providing an algorithm of running time $2^{O(k^2)} γn^6 \log n$, where $γ$ is the time to compute $f(X)$ for any set $X$, and $n = |V|$. This improves the previous algorithm by Oum and Seymour [J. Combin. Theory Ser. B, 2007], which runs in time $γn^{O(k)}$. Our algorithm can be applied to rank-width of graphs, branch-width of matroids, branch-width of (hyper)graphs, and carving-width of graphs. This resolves an open problem asked by Hliněný [SIAM J. Comput., 2005], who asked whether branch-width of matroids given by the rank oracle is fixed-parameter tractable. Furthermore, our algorithm improves the best known dependency on $k$ in the running times of FPT algorithms for graph branch-width, rank-width, and carving-width.
- Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions (with Marek Sokołowski), In the Proceedings of the 34th Annual European Symposium on Algorithms (ESA 2026, L’Aquila, Italy, August 31-September 4, 2026), accepted, August 2026.We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining $0$ on the empty set. For a connectivity function $f$ on an $n$-element set $V$ and an integer $k\ge 0$, we show that the family of all sets $X\subseteq V$ with $f(X)=k$ admits a polynomial-size representation: it can be described by a list of at most $O(n^{4k})$ items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets $X$ with $f(X)=k$. We also give an algorithm that constructs this representation in time $O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})$, where $γ$ is the oracle time to evaluate $f$. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, arXiv, 2025] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed $k$, we obtain a polynomial-time algorithm for finding a set $A$ with $f(A)=k$ and a prescribed cardinality constraint on $A$.
Polynomial-size encoding of all cuts of small value in integer-valued symmetric submodular functions
We study connectivity functions, that is, integer-valued symmetric submodular functions on a finite ground set attaining $0$ on the empty set. For a connectivity function $f$ on an $n$-element set $V$ and an integer $k\ge 0$, we show that the family of all sets $X\subseteq V$ with $f(X)=k$ admits a polynomial-size representation: it can be described by a list of at most $O(n^{4k})$ items, each consisting of a set to be included, another set to be excluded, and a partition of remaining elements, such that the union of some members of the partition and the set to be included are precisely all sets $X$ with $f(X)=k$. We also give an algorithm that constructs this representation in time $O(n^{2k+7}γ+n^{2k+8}+n^{4k+2})$, where $γ$ is the oracle time to evaluate $f$. This generalizes the low rank structure theorem of Bojańczyk, Pilipczuk, Przybyszewski, Sokołowski, and Stamoulis [Low rank MSO, arXiv, 2025] on cut-rank functions on graphs to general connectivity functions. As an application, for fixed $k$, we obtain a polynomial-time algorithm for finding a set $A$ with $f(A)=k$ and a prescribed cardinality constraint on $A$.
- The Erdős-Pósa property for circle graphs as vertex-minors (with Rutger Campbell, J. Pascal Gollin, Meike Hatzel, O-joung Kwon권오정, Rose McCarty, and Sebastian Wiederrecht), In the Proceedings of the Thirty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2026, January 11-14, 2026, Vancouver, Canada), pp. 4930-4952, January 2026.We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$.
The Erdős-Pósa property for circle graphs as vertex-minors
We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$.
2025
- Recognisability Equals Definability for Finitely Representable Matroids of Bounded Path-Width (with Mamadou Moustapha Kanté, Bruno Guillon, Eun Jung Kim김은정, and Rutger Campbell), In the Proceedings of the 40th Annual ACM-IEEE Symposium on Logic in Computer Science (LICS 2025, Singapore, June 23-26, 2025), pp. 678-690, October 2025.Let ${\mathbb{F}}$ be a finite field. We prove that there is an MSO-transduction which, given an ${\mathbb{F}}$-representable matroid of path-width k, produces a branch-decomposition of width at most f(k), for some function f. As a corollary, any recognizable property of ${\mathbb{F}}$-representable matroids with bounded path-width is definable in MSO logic, and therefore recognizability is equivalent to MSO-definability on classes of ${\mathbb{F}}$-representable matroids of bounded path-width. This generalizes the result of Bojańczyk, Grohe and Pilipczuk [Logical Methods in Computer Science 17(1), 2021] which asserts the equivalence of the two notions on graphs of bounded linear clique-width.
Recognisability Equals Definability for Finitely Representable Matroids of Bounded Path-Width
Let ${\mathbb{F}}$ be a finite field. We prove that there is an MSO-transduction which, given an ${\mathbb{F}}$-representable matroid of path-width k, produces a branch-decomposition of width at most f(k), for some function f. As a corollary, any recognizable property of ${\mathbb{F}}$-representable matroids with bounded path-width is definable in MSO logic, and therefore recognizability is equivalent to MSO-definability on classes of ${\mathbb{F}}$-representable matroids of bounded path-width. This generalizes the result of Bojańczyk, Grohe and Pilipczuk [Logical Methods in Computer Science 17(1), 2021] which asserts the equivalence of the two notions on graphs of bounded linear clique-width.
2023
- Space-efficient parameterized algorithms on graphs of low shrubdepth (with Vera Chekan, Robert Ganian, Mamadou Moustapha Kanté, Michał Pilipczuk, Erik Jan van Leeuwen, Benjamin Bergougnoux, and Matthias Mnich), In the Proceedings of the 31st Annual European Symposium on Algorithms (ESA 2023, Amsterdam, Netherlands, September 4-6, 2023), Article No. 18; pp. 18:1-18:8, September 2023.Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. \nMotivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, \n- Independent Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using 𝒪(dk²log n) space; \n- Max Cut can be solved in time n^𝒪(dk) using 𝒪(dk log n) space; and \n- Dominating Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using n^𝒪(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial.
Space-efficient parameterized algorithms on graphs of low shrubdepth
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. \nMotivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, \n- Independent Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using 𝒪(dk²log n) space; \n- Max Cut can be solved in time n^𝒪(dk) using 𝒪(dk log n) space; and \n- Dominating Set can be solved in time 2^𝒪(dk) ⋅ n^𝒪(1) using n^𝒪(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial.
2022
- Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k (with Mamadou Moustapha Kanté, Eun Jung Kim김은정, and O-joung Kwon권오정), In the Proceedings of the 39th International Symposium on Theoretical Aspects of Computer Science (STACS 2022, Marseille, March 15-18, 2022), Article No. 40; pp. 40:1-40:14, March 2022.Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field F, the list contains only finitely many F-representable matroids, due to the well-quasi-ordering of F-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these F-representable excluded minors in general. \nWe consider the class of matroids of path-width at most k for fixed k. We prove that for a finite field F, every F-representable excluded minor for the class of matroids of path-width at most k has at most 2^{|𝔽|^{O(k²)}} elements. We can therefore compute, for any integer k and a fixed finite field F, the set of F-representable excluded minors for the class of matroids of path-width k, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an F-represented matroid is at most k. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most k has at most 2^{2^{O(k²)}} vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.
Obstructions for matroids of path-width at most k and graphs of linear rank-width at most k
Every minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field F, the list contains only finitely many F-representable matroids, due to the well-quasi-ordering of F-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these F-representable excluded minors in general. \nWe consider the class of matroids of path-width at most k for fixed k. We prove that for a finite field F, every F-representable excluded minor for the class of matroids of path-width at most k has at most 2^{|𝔽|^{O(k²)}} elements. We can therefore compute, for any integer k and a fixed finite field F, the set of F-representable excluded minors for the class of matroids of path-width k, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an F-represented matroid is at most k. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most k has at most 2^{2^{O(k²)}} vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs.
2021
- Γ-graphic delta-matroids and their applications (with Donggyu Kim김동규 and Duksang Lee이덕상), In the Proceedings of the 32nd International Symposium on Algorithms and Computation (ISAAC2021, December 6-8, 2021, Fukuoka, Japan), Article No. 70; pp. 70:1-70:13, December 2021.For an abelian group $Γ$, a $Γ$-labelled graph is a graph whose vertices are labelled by elements of $Γ$. We prove that a certain collection of edge sets of a $Γ$-labelled graph forms a delta-matroid, which we call a $Γ$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by $k$ and Maximum Weight $S$-Tree Packing. We also discuss various properties of $Γ$-graphic delta-matroids.
Γ-graphic delta-matroids and their applications
For an abelian group $Γ$, a $Γ$-labelled graph is a graph whose vertices are labelled by elements of $Γ$. We prove that a certain collection of edge sets of a $Γ$-labelled graph forms a delta-matroid, which we call a $Γ$-graphic delta-matroid, and provide a polynomial-time algorithm to solve the separation problem, which allows us to apply the symmetric greedy algorithm of Bouchet to find a maximum weight feasible set in such a delta-matroid. We present two algorithmic applications on graphs; Maximum Weight Packing of Trees of Order Not Divisible by $k$ and Maximum Weight $S$-Tree Packing. We also discuss various properties of $Γ$-graphic delta-matroids.
2020
- A polynomial kernel for 3-leaf power deletion (with Jungho Ahn안정호, Eduard Eiben, and O-joung Kwon권오정), In the Proceedings of the 45th International Symposium on Mathematical Foundations of Computer Science (MFCS2020, August 24-28, 2020, Prague, Czech Republic), Article No. 5; pp. 5:1-5:14, August 2020.For a non-negative integer 𝓁, a graph G is an 𝓁-leaf power of a tree T if V(G) is equal to the set of leaves of T, and distinct vertices v and w of G are adjacent if and only if the distance between v and w in T is at most 𝓁. Given a graph G, 3-Leaf Power Deletion asks whether there is a set S ⊆ V(G) of size at most k such that G\\S is a 3-leaf power of some treeT. We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance (G,k) to output an equivalent instance (G',k') such that k'≤ k and G' has at most O(k^14) vertices.
A polynomial kernel for 3-leaf power deletion
For a non-negative integer 𝓁, a graph G is an 𝓁-leaf power of a tree T if V(G) is equal to the set of leaves of T, and distinct vertices v and w of G are adjacent if and only if the distance between v and w in T is at most 𝓁. Given a graph G, 3-Leaf Power Deletion asks whether there is a set S ⊆ V(G) of size at most k such that G\\S is a 3-leaf power of some treeT. We provide a polynomial kernel for this problem. More specifically, we present a polynomial-time algorithm for an input instance (G,k) to output an equivalent instance (G',k') such that k'≤ k and G' has at most O(k^14) vertices.
2018
- Finding branch-decompositions of matroids, hypergraphs, and more (with Jisu Jeong정지수 and Eun Jung Kim김은정), In the Proceedings of the 45th International Colloquium on Automata, Languages, and Programming (ICALP2018), Prague, Czech Republic, July 9-13, 2018, Leibniz International Proceedings in Informatics (LIPIcs), Vol. 107, pp. 80:1-80:14, July 2018.Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 6 September 2019Accepted: 03 July 2021Published online: 04 November 2021Keywordsbranch-width, rank-width, carving-width, matroid, fixed-parameter tractabilityAMS Subject Headings68Q25, 68W40, 05C50Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec
Finding branch-decompositions of matroids, hypergraphs, and more
Related DatabasesWeb of Science You must be logged in with an active subscription to view this.Article DataHistorySubmitted: 6 September 2019Accepted: 03 July 2021Published online: 04 November 2021Keywordsbranch-width, rank-width, carving-width, matroid, fixed-parameter tractabilityAMS Subject Headings68Q25, 68W40, 05C50Publication DataISSN (print): 0895-4801ISSN (online): 1095-7146Publisher: Society for Industrial and Applied MathematicsCODEN: sjdmec
- Computing small pivot-minors (with Konrad K. Dabrowski, Jisu Jeong정지수, Mamadou Moustapha Kanté, O-joung Kwon권오정, Daniël Paulusma, and François Dross), In the Proceedings of the 44th International Workshop on Graph-Theoretic Concepts in Computer Science (WG2018, Cottbus, Germany, June 27-29, 2018), Lecture Notes in Comput. Sci., vol 11159, pp. 125-138, June 2018.
2016
- An FPT 2-approximation for tree-cut decomposition (with Eun Jung Kim김은정, Christophe Paul, Ignasi Sau, and Dimitrios M. Thilikos), In the Proceedings of the 13th Workshop on Approximation and Online Algorithms (WAOA 2015), pp 35-46, 2016.The tree-cut width of a graph is a graph parameter defined by Wollan [J. Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized 2-approximation algorithm for the computation of tree-cut width; for an input $n$-vertex graph $G$ and an integer $w$, our algorithm either confirms that the tree-cut width of $G$ is more than $w$ or returns a tree-cut decomposition of $G$ certifying that its tree-cut width is at most $2w$, in time $2^{O(w^2\log w)} \cdot n^2$. Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known.
An FPT 2-approximation for tree-cut decomposition
The tree-cut width of a graph is a graph parameter defined by Wollan [J. Comb. Theory, Ser. B, 110:47-66, 2015] with the help of tree-cut decompositions. In certain cases, tree-cut width appears to be more adequate than treewidth as an invariant that, when bounded, can accelerate the resolution of intractable problems. While designing algorithms for problems with bounded tree-cut width, it is important to have a parametrically tractable way to compute the exact value of this parameter or, at least, some constant approximation of it. In this paper we give a parameterized 2-approximation algorithm for the computation of tree-cut width; for an input $n$-vertex graph $G$ and an integer $w$, our algorithm either confirms that the tree-cut width of $G$ is more than $w$ or returns a tree-cut decomposition of $G$ certifying that its tree-cut width is at most $2w$, in time $2^{O(w^2\log w)} \cdot n^2$. Prior to this work, no constructive parameterized algorithms, even approximated ones, existed for computing the tree-cut width of a graph. As a consequence of the Graph Minors series by Robertson and Seymour, only the existence of a decision algorithm was known.
- Constructive algorithm for path-width of matroids (with Jisu Jeong정지수 and Eun Jung Kim김은정), In the Proceedings of the 27th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2016, Arlington, VA, 2016). pp. 1695-1704, 2016.Given n subspaces of a finite-dimensional vector space over a fixed finite field F, we wish to find a linear layout V1, V2, …, Vn of the subspaces such that dim((V1 + V2 + ⃛ + Vi)∩(Vi+1 + ⃛ + Vn)) ≤ k for all i; such a linear layout is said to have width at most k. When restricted to 1-dimensional subspaces, this problem is equivalent to computing the path-width of an F-represented matroid in matroid theory and computing the trellis-width (or minimum trellis state-complexity) of a linear code in coding theory. We present a fixed-parameter tractable algorithm to construct a linear layout of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over F. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most k for an input F-represented matroid of path-width at most k, and a fixed-parameter tractable algorithm to find a linear rank-decomposition of width at most k for an input graph of linear rank-width at most k. In both corollaries, no such algorithms were known previously. Our approach is based on dynamic programming combined with the idea developed by Bodlaender and Kloks (1996) for their work on path-width and tree-width of graphs. It was previously known that a fixed-parameter tractable algorithm exists for the decision version of the problem for matroid path-width; a theorem by Geelen, Gerards, and Whittle (2002) implies that for each fixed finite field F, there are finitely many forbidden F-representable minors for the class of matroids of path-width at most k. An algorithm by Hliněný (2006) can detect a minor in an input F-represented matroid of bounded branch-width. However, this indirect approach would not produce an actual path-decomposition even if the complete list of forbidden minors were known. Our algorithm is the first one to construct such a path-decomposition and does not depend on the finiteness of forbidden minors.
Constructive algorithm for path-width of matroids
Given n subspaces of a finite-dimensional vector space over a fixed finite field F, we wish to find a linear layout V1, V2, …, Vn of the subspaces such that dim((V1 + V2 + ⃛ + Vi)∩(Vi+1 + ⃛ + Vn)) ≤ k for all i; such a linear layout is said to have width at most k. When restricted to 1-dimensional subspaces, this problem is equivalent to computing the path-width of an F-represented matroid in matroid theory and computing the trellis-width (or minimum trellis state-complexity) of a linear code in coding theory. We present a fixed-parameter tractable algorithm to construct a linear layout of width at most k, if it exists, for input subspaces of a finite-dimensional vector space over F. As corollaries, we obtain a fixed-parameter tractable algorithm to produce a path-decomposition of width at most k for an input F-represented matroid of path-width at most k, and a fixed-parameter tractable algorithm to find a linear rank-decomposition of width at most k for an input graph of linear rank-width at most k. In both corollaries, no such algorithms were known previously. Our approach is based on dynamic programming combined with the idea developed by Bodlaender and Kloks (1996) for their work on path-width and tree-width of graphs. It was previously known that a fixed-parameter tractable algorithm exists for the decision version of the problem for matroid path-width; a theorem by Geelen, Gerards, and Whittle (2002) implies that for each fixed finite field F, there are finitely many forbidden F-representable minors for the class of matroids of path-width at most k. An algorithm by Hliněný (2006) can detect a minor in an input F-represented matroid of bounded branch-width. However, this indirect approach would not produce an actual path-decomposition even if the complete list of forbidden minors were known. Our algorithm is the first one to construct such a path-decomposition and does not depend on the finiteness of forbidden minors.
2013
- Excluded vertex-minors for graphs of linear rank-width at most k (with Jisu Jeong정지수 and O-joung Kwon권오정), In the Proceedings of the 30th Symposium on Theoretical Aspects of Computer Science (STACS’13), Kiel, Germany, Feb. 27-Mar. 02, 2013, Leibniz International Proceedings in Informatics (LIPIcs), Vol. 20, pp. 221-232, 2013.Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each k, there is a finite set \mathcal{O}_k of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomorphic to a graph in \mathcal{O}_k. However, no attempts have been made to bound the number of graphs in \mathcal{O}_k for k >= 2. We construct, for each k, 2^{\Omega(3^k)} pairwise locally non-equivalent graphs that are excluded vertex-minors for graphs of linear rank-width at most k. Therefore the number of graphs in \mathcal{O}_k is at least double exponential.
Excluded vertex-minors for graphs of linear rank-width at most k
Linear rank-width is a graph width parameter, which is a variation of rank-width by restricting its tree to a caterpillar. As a corollary of known theorems, for each k, there is a finite set \mathcal{O}_k of graphs such that a graph G has linear rank-width at most k if and only if no vertex-minor of G is isomorphic to a graph in \mathcal{O}_k. However, no attempts have been made to bound the number of graphs in \mathcal{O}_k for k >= 2. We construct, for each k, 2^{\Omega(3^k)} pairwise locally non-equivalent graphs that are excluded vertex-minors for graphs of linear rank-width at most k. Therefore the number of graphs in \mathcal{O}_k is at least double exponential.
2012
- Deciding first order logic properties of matroids (with Tomáš Gavenčiak and Daniel Kráľ), In the Proceedings of the 39th International Colloquium on Automata, Languages, and Programming (ICALP 2012), Warwick, UK, July 9-13, 2012, Part II, Lecture Notes in Comput. Sci., Vol. 7392, pp. 239-250, July 2012.Frick and Grohe [J. ACM 48 (2006), 1184-1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order logic property can be decided in almost linear time in such a graph class. Here, we introduce an analogous notion for matroids (locally bounded branch-width) and show the existence of a fixed parameter algorithm for first order logic properties in classes of regular matroids with locally bounded branch-width. To obtain this result, we show that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.
Deciding first order logic properties of matroids
Frick and Grohe [J. ACM 48 (2006), 1184-1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order logic property can be decided in almost linear time in such a graph class. Here, we introduce an analogous notion for matroids (locally bounded branch-width) and show the existence of a fixed parameter algorithm for first order logic properties in classes of regular matroids with locally bounded branch-width. To obtain this result, we show that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.
2007
- Finding Branch-decompositions and Rank-decompositions (with Petr Hliněný), In the Proceedings of the 15th Annual European Symposium on Algorithms (ESA 2007, Eilat, Israel, October 8-10, 2007), Lecture Notes in Comput. Sci., vol 4698, pp. 163-174, October 2007.We present a new algorithm that can output the rank- decomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixed-parameter tractable, that is, they run in time O(n3 ) for each fixed value of k where n is the number of vertices / elements of the input. (The previous best algorithm for construction of a branch- decomposition or a rank-decomposition of optimal width due to Oum and Seymour [Testing branch-width. J. Combin. Theory Ser. B, 97(3) (2007) 385–393] is not fixed-parameter tractable.)
Finding Branch-decompositions and Rank-decompositions
We present a new algorithm that can output the rank- decomposition of width at most k of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most k if such exists. This algorithm works also for partitioned matroids. Both these algorithms are fixed-parameter tractable, that is, they run in time O(n3 ) for each fixed value of k where n is the number of vertices / elements of the input. (The previous best algorithm for construction of a branch- decomposition or a rank-decomposition of optimal width due to Oum and Seymour [Testing branch-width. J. Combin. Theory Ser. B, 97(3) (2007) 385–393] is not fixed-parameter tractable.)
2006
- Certifying Large Branch-width (with Paul Seymour), In the Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms (Miami, FL, 2006). SODA ’06. ACM, New York, pp. 810-813, 2006.We are not yet able to find a polynomial-time Branch-width is defined for graphs, matroids, and, more algorithm to decide whether branch-width is at most k, generally, arbitrary symmetric submodular functions. For but in [5], we give a polynomial-time “approximation” a finite set V, a function f on the set of subsets 2V of V algorithm that, for fixed k, either confirms that branch- is submodular if f (X) + f (Y ) ≥ f (X ∩ Y ) + f (X ∪ Y ), width of a connectivity function is larger than k or and symmetric if f (X) = f (V \ X). We discuss the obtains a branch-decomposition of width at most 3k +1. computational complexity of recognizing that symmetric There have been answers for our problem for a few submodular functions have branch-width at most k for fixed special symmetric submodular functions separately. We k. An integer-valued symmetric submodular function f on summarize them in Table 1. In particular, it is open 2V is a connectivity function if f (∅) = 0 and f ({v}) ≤ 1 whether there exists a polynomial-time algorithm that for all v ∈ V. We show that for each constant k, if a decides whether a matroid (given by an independence connectivity function f on 2V is presented by an oracle oracle) has branch-width at most k for fixed k. More- and the branch-width of f is larger than k, then there is a over, this problem is open when the input matroid is certificate of polynomial size (in |V |) such that a polynomial- represented over a fixed non-finite field. Our result im- time algorithm can verify the claim that branch-width of f plies that it is in NP∩coNP to decide that branch-width is larger than k. In particular it is in coNP to recognize of represented matroids is at most k; in this case we matroids represented over a fixed field with branch-width at do not need an oracle to obtain the input matroid and most k for fixed k. therefore we can say that our algorithm is in coNP.
Certifying Large Branch-width
We are not yet able to find a polynomial-time Branch-width is defined for graphs, matroids, and, more algorithm to decide whether branch-width is at most k, generally, arbitrary symmetric submodular functions. For but in [5], we give a polynomial-time “approximation” a finite set V, a function f on the set of subsets 2V of V algorithm that, for fixed k, either confirms that branch- is submodular if f (X) + f (Y ) ≥ f (X ∩ Y ) + f (X ∪ Y ), width of a connectivity function is larger than k or and symmetric if f (X) = f (V \ X). We discuss the obtains a branch-decomposition of width at most 3k +1. computational complexity of recognizing that symmetric There have been answers for our problem for a few submodular functions have branch-width at most k for fixed special symmetric submodular functions separately. We k. An integer-valued symmetric submodular function f on summarize them in Table 1. In particular, it is open 2V is a connectivity function if f (∅) = 0 and f ({v}) ≤ 1 whether there exists a polynomial-time algorithm that for all v ∈ V. We show that for each constant k, if a decides whether a matroid (given by an independence connectivity function f on 2V is presented by an oracle oracle) has branch-width at most k for fixed k. More- and the branch-width of f is larger than k, then there is a over, this problem is open when the input matroid is certificate of polynomial size (in |V |) such that a polynomial- represented over a fixed non-finite field. Our result im- time algorithm can verify the claim that branch-width of f plies that it is in NP∩coNP to decide that branch-width is larger than k. In particular it is in coNP to recognize of represented matroids is at most k; in this case we matroids represented over a fixed field with branch-width at do not need an oracle to obtain the input matroid and most k for fixed k. therefore we can say that our algorithm is in coNP.
2005
- Approximating Rank-width and Clique-width Quickly, 31st International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Comput. Sci., vol 3787, pp. 49-58, 2005.Rank-width was defined by Oum and Seymour [2006] to investigate clique-width. They constructed an algorithm that either outputs a rank-decomposition of width at most f ( k ) for some function f or confirms that rank-width is larger than k in time O (| V | 9 log | V |) for an input graph G = ( V, E ) and a fixed k. We develop three separate algorithms of this kind with faster running time. We construct an O (| V | 4 )-time algorithm with f ( k ) = 3 k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O (| V | 3 )-time algorithm with f ( k ) = 24 k, achieved by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný [2005]. Finally we construct an O (| V | 3 )-time algorithm with f ( k ) = 3 k − 1 by combining the ideas of the two previously cited papers.
Approximating Rank-width and Clique-width Quickly
Rank-width was defined by Oum and Seymour [2006] to investigate clique-width. They constructed an algorithm that either outputs a rank-decomposition of width at most f ( k ) for some function f or confirms that rank-width is larger than k in time O (| V | 9 log | V |) for an input graph G = ( V, E ) and a fixed k. We develop three separate algorithms of this kind with faster running time. We construct an O (| V | 4 )-time algorithm with f ( k ) = 3 k + 1 by constructing a subroutine for the previous algorithm; we avoid generic algorithms minimizing submodular functions used by Oum and Seymour. Another one is an O (| V | 3 )-time algorithm with f ( k ) = 24 k, achieved by giving a reduction from graphs to binary matroids; then we use an approximation algorithm for matroid branch-width by Hliněný [2005]. Finally we construct an O (| V | 3 )-time algorithm with f ( k ) = 3 k − 1 by combining the ideas of the two previously cited papers.
Other
- Unifying duality theorems for width parameters in graphs and matroids (with Reinhard Diestel), 40th International Workshop on Graph-Theoretic Concepts in Computer Science, Lecture Notes in Comput. Sci., vol 8747, pp. 1-14, 2014.
- Branch-width and Tangles (with Illya V. Hicks), Wiley Encyclopedia of Operations Research and Management Sciences, 2011.This article describes the notion of branch‐width and its dual notion, tangles. Branch‐width was introduced by Robertson and Seymour and has been applied to various combinatorial structures.
Branch-width and Tangles
This article describes the notion of branch‐width and its dual notion, tangles. Branch‐width was introduced by Robertson and Seymour and has been applied to various combinatorial structures.
Manuscripts
- An upper bound on tricolored ordered sum-free sets (with Taegyun Kim김태균), 2017.We present a strengthening of the lemma on the lower bound of the slice rank by Tao (2016) motivated by the Croot-Lev-Pach-Ellenberg-Gijswijt bound on cap sets (2017, 2017). The Croot-Lev-Pach-Ellenberg-Gijswijt method and the lemma of Tao are based on the fact that the rank of a diagonal matrix is equal to the number of non-zero diagonal entries. Our lemma is based on the rank of upper-triangular matrices. This stronger lemma allows us to prove the following extension of the Ellenberg-Gijswijt result (2017). A tricolored ordered sum-free set in $\mathbb F_p^n$ is a collection $\{(a_i,b_i,c_i):i=1,2,\ldots,m\}$ of ordered triples in $(\mathbb F_p^n )^3$ such that $a_i+b_i+c_i=0$ and if $a_i+b_j+c_k=0$, then $i\le j\le k$. By using the new lemma, we present an upper bound on the size of a tricolored ordered sum-free set in $\mathbb F_p^n$.
An upper bound on tricolored ordered sum-free sets
We present a strengthening of the lemma on the lower bound of the slice rank by Tao (2016) motivated by the Croot-Lev-Pach-Ellenberg-Gijswijt bound on cap sets (2017, 2017). The Croot-Lev-Pach-Ellenberg-Gijswijt method and the lemma of Tao are based on the fact that the rank of a diagonal matrix is equal to the number of non-zero diagonal entries. Our lemma is based on the rank of upper-triangular matrices. This stronger lemma allows us to prove the following extension of the Ellenberg-Gijswijt result (2017). A tricolored ordered sum-free set in $\mathbb F_p^n$ is a collection $\{(a_i,b_i,c_i):i=1,2,\ldots,m\}$ of ordered triples in $(\mathbb F_p^n )^3$ such that $a_i+b_i+c_i=0$ and if $a_i+b_j+c_k=0$, then $i\le j\le k$. By using the new lemma, we present an upper bound on the size of a tricolored ordered sum-free set in $\mathbb F_p^n$.
- Unifying duality theorems for width parameters in graphs and matroids. II. General duality (with Reinhard Diestel), 2014.We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]
Unifying duality theorems for width parameters in graphs and matroids. II. General duality
We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6]
- Frick and Grohe [J. ACM 48 (2006), 1184-1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order logic property can be decided in almost linear time in such a graph class. Here, we introduce an analogous notion for matroids (locally bounded branch-width) and show the existence of a fixed parameter algorithm for first order logic properties in classes of regular matroids with locally bounded branch-width. To obtain this result, we show that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.
Deciding first order logic properties of matroids
Frick and Grohe [J. ACM 48 (2006), 1184-1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order logic property can be decided in almost linear time in such a graph class. Here, we introduce an analogous notion for matroids (locally bounded branch-width) and show the existence of a fixed parameter algorithm for first order logic properties in classes of regular matroids with locally bounded branch-width. To obtain this result, we show that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.
- Injective Chromatic Number and Chromatic Number of the Square of Graphs (with Seog-Jin Kim김석진), manuscript, 2009., 2011.The injective chromatic number of a graph G is the minimum number of colors needed in order to color vertices of G so that two vertices with a common neighbor receive distinct colors. We prove that the injective chromatic number of G is at least the half of the chromatic number of G 2, the square of G. This inequality is tight. An injective k-coloring of a graph G is an assignment of at most k colors to the vertices of G such that two vertices sharing a common neighbor must have distinct colors. The injective chromatic number i(G) of a graph G is the minimum k such that G has an injective k-coloring. This notion was
Injective Chromatic Number and Chromatic Number of the Square of Graphs
The injective chromatic number of a graph G is the minimum number of colors needed in order to color vertices of G so that two vertices with a common neighbor receive distinct colors. We prove that the injective chromatic number of G is at least the half of the chromatic number of G 2, the square of G. This inequality is tight. An injective k-coloring of a graph G is an assignment of at most k colors to the vertices of G such that two vertices sharing a common neighbor must have distinct colors. The injective chromatic number i(G) of a graph G is the minimum k such that G has an injective k-coloring. This notion was
AMS MathSciNet Author ID: 765385
Scopus Author ID: 55912957400
ResearchID: C-1692-2011
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