On the tree-width of even-hole-free graphs

Aboulker, P; Adler, Isolde; Kim, Eun Jung; Sintiari, Ni Luh Dewi; Trotignon, Nicolas

doi:10.48550/arxiv.2008.05504

Cited by 4 publications

(21 citation statements)

References 16 publications

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“…Let c ∈ [ 1 2 , 1) and let G be a graph. If G has a (w, c)-balanced separator of size k for every uniform weight function w, then tw(G) ≤ 1 1−c k. Lemma 1.13 implies that if for some fixed c ∈ [ 1 2 , 1), G has a balanced separator of size k for every weight function w, then the treewidth of G is bounded by a function of k. The next lemma states the converse. Lemma 1.14 ([7]).…”

Section: Theorem 17 ([1]mentioning

confidence: 95%

“…Observing that graphs G in Theorem 1.4 have vertices of arbitrarily large degree, the following conjecture was made (and proved for the case ∆ ≤ 3) in [1]: Conjecture 1.5. For every ∆ > 0 there exists c ∆ such that even-hole-free graphs with maximum degree at most ∆ have treewidth at most c ∆ .…”

Section: Theorem 11 ([12]mentioning

confidence: 99%

“…Conjecture 1.5 was proved in [3] by three of the authors of the present paper. More generally, it is conjectured in [1] that there is an affirmative asnwer to Questoin 1.3 in the bounded maximum degree case (note that bounded maximum degree automatically implies that a complete graph and a complete bipartite graph is excluded).…”

Section: Theorem 11 ([12]mentioning

confidence: 99%

“…This remains open, though in [1] it is proved for proper minor-closed classes of graphs (in which case the bound on the maximum degree is not needed anymore).…”

Section: Theorem 11 ([12]mentioning

confidence: 99%

See 3 more Smart Citations

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

Abrishami¹,

Chudnovsky²,

Dibek³

et al. 2021

Preprint

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This paper is motivated by the following question: what are the unavoidable induced subgraphs of graphs with large treewidth? Aboulker et al. made a conjecture which answers this question in graphs of bounded maximum degree, asserting that for all k and ∆, every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a subdivision of the (k × k)-wall or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. We prove two theorems supporting this conjecture, as follows.1. For t ≥ 2, a t-theta is a graph consisting of two nonadjacent vertices and three internally disjoint paths between them, each of length at least t. A t-pyramid is a graph consisting of a vertex v, a triangle B disjoint from v and three paths starting at v and disjoint otherwise, each joining v to a vertex of B, and each of length at least t. We prove that for all k, t and ∆, every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a t-theta, or a t-pyramid, or the line graph of a subdivision of the (k × k)-wall as an induced subgraph. This affirmatively answers a question of Pilipczuk et al. asking whether every graph of bounded maximum degree and sufficiently large treewidth contains either a theta or a triangle as an induced subgraph (where a theta means a t-theta for some t ≥ 2). 2. A subcubic subdivided caterpillar is a tree of maximum degree at most three whose all vertices of degree three lie on a path. We prove that for every ∆ and subcubic subdivided caterpillar T , every graph with maximum degree at most ∆ and sufficiently large treewidth contains either a subdivision of T or the line graph of a subdivision of T as an induced subgraph.

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Section: Theorem 17 ([1]mentioning

confidence: 95%

Section: Theorem 11 ([12]mentioning

confidence: 99%

Section: Theorem 11 ([12]mentioning

confidence: 99%

“…This remains open, though in [1] it is proved for proper minor-closed classes of graphs (in which case the bound on the maximum degree is not needed anymore).…”

Section: Theorem 11 ([12]mentioning

confidence: 99%

See 2 more Smart Citations

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

Abrishami¹,

Chudnovsky²,

Dibek³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…In [2], with Dibek, Rzążewski and Vušković, we gave an affirmative answer to this question. More generally, it is also conjectured in [1] that there is an affirmative answer to Question 1.2 restricted to graphs of bounded maximum degree.…”

Section: Theorem 11 ([15]mentioning

confidence: 99%

Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

Abrishami¹,

Chudnovsky²,

Hajebi³

et al. 2021

Preprint

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A theta is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family H of graphs, we say a graph G is H-free if no induced subgraph of G is isomorphic to a member of H. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant c for which every (theta, triangle)-free graph G has treewidth at most c log(|V (G)|). A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth.Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in |V (G)| for every graph G excluding the so-called three-path-configurations as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and k-Coloring (for fixed k) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.

show abstract

Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

Abrishami

Chudnovsky

Hajebi

et al. 2022

AiC

View full text Add to dashboard Cite

A _theta_ is a graph consisting of two non-adjacent vertices and three internally disjoint paths between them, each of length at least two. For a family $\mathcal{H}$ of graphs, we say a graph $G$ is $\mathcal{H}$-_free_ if no induced subgraph of $G$ is isomorphic to a member of $\mathcal{H}$. We prove a conjecture of Sintiari and Trotignon, that there exists an absolute constant $c$ for which every (theta, triangle)-free graph $G$ has treewidth at most $c\log (|V(G)|)$. A construction by Sintiari and Trotignon shows that this bound is asymptotically best possible, and (theta, triangle)-free graphs comprise the first known hereditary class of graphs with arbitrarily large yet logarithmic treewidth. Our main result is in fact a generalization of the above conjecture, that treewidth is at most logarithmic in $|V(G)|$ for every graph $G$ excluding the so-called _three-path-configurations_ as well as a fixed complete graph. It follows that several NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and $k$-Coloring (for fixed $k$) admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.

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On the tree-width of even-hole-free graphs

Cited by 4 publications

References 16 publications

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

Induced subgraphs and tree decompositions II. Toward walls and their line graphs in graphs of bounded degree

Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

Induced subgraphs and tree decompositions III. Three-path-configurations and logarithmic treewidth

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