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

Abrishami, Tara; Chudnovsky, Maria; Dibek, Cemil; Hajebi, Sepehr; Rzążewski, Paweł; Spirkl, Sophie; Vušković, Kristina

doi:10.48550/arxiv.2108.01162

Cited by 9 publications

(24 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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: 80%

“…This is still open, while several interesting special cases of it are proved in earlier papers of this series [2,4]. In the same vein, the following may be true as far as we know (this is a variant of a conjecture of [18]): Conjecture 1.9.…”

Section: Conjecture 18 ([1]mentioning

confidence: 81%

“…The proof of Theorem 1.5 builds on a method developed in two previous papers of this series [2,4] to bound the treewidth of graph classes with bounded maximum degree. Incidentally, unlike subdivided walls and their line graphs, the graphs G ℓ from Theorem 1.3 contain vertices of arbitrarily large degree.…”

Section: Theorem 11 ([15]mentioning

confidence: 99%

See 2 more Smart Citations

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

Abrishami¹,

Chudnovsky²,

Hajebi³

et al. 2021

Preprint

Self Cite

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 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

Section: Theorem 11 ([15]mentioning

confidence: 80%

Section: Conjecture 18 ([1]mentioning

confidence: 81%

See 1 more Smart Citation

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

Abrishami¹,

Chudnovsky²,

Hajebi³

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

show abstract

“…We start with the following lemma (see Lemma 18 in [4] for a variation of this lemma). This proof of Lemma 5.1 originally appeared in [1], but we include it here for completeness. Lemma 5.1.…”

Section: Forcers For (C 4 Prism)-free Perfect Graphsmentioning

confidence: 99%

Submodular functions and perfect graphs

Abrishami,

Chudnovsky,

Dibek

et al. 2021

Preprint

Self Cite

View full text Add to dashboard Cite

We give a combinatorial polynomial-time algorithm to find a maximum weight independent set in perfect graphs of bounded degree that do not contain a prism or a hole of length four as an induced subgraph. An even pair in a graph is a pair of vertices all induced paths between which are even. An even set is a set of vertices every two of which are an even pair. We show that every perfect graph that does not contain a prism or a hole of length four as an induced subgraph has a balanced separator which is the union of a bounded number of even sets, where the bound depends only on the maximum degree of the graph. This allows us to solve the maximum weight independent set problem using the well-known submodular function minimization algorithm.

show abstract

“…Robertson and Seymour famously gave a complete answer to this question in the case of subgraphs. By W k×k we denote the (k × k)-wall; see [2] for a full definition.…”

Section: Introductionmentioning

confidence: 99%

Induced subgraphs and tree decompositions IV. (Even hole, diamond, pyramid)-free graphs

Abrishami¹,

Chudnovsky²,

Hajebi³

et al. 2022

Preprint

Self Cite

View full text Add to dashboard Cite

A hole in a graph G is an induced cycle of length at least four, and an even hole is a hole of even length. The diamond is the graph obtained from the complete graph K4 by removing an edge. A pyramid is a graph consisting of a triangle called the base, a vertex called the apex, and three internally disjoint paths starting at the apex and disjoint otherwise, each joining the apex to a vertex of the base. For a family H of graphs, we say a graph G is Hfree if no induced subgraph of G is isomorphic to a member of H. Cameron, da Silva, Huang, and Vušković proved that (even hole, triangle)-free graphs have treewidth at most five, which motivates studying the treewidth of even-hole-free graphs of larger clique number. Sintiari and Trotignon provided a construction of (even hole, K4)-free graphs of arbitrarily large treewidth, and observing induced diamonds all over their construction, they conjectured that (even hole, diamond, K4)-free graphs have bounded treewidth. We approach this conjecture by showing that for every t, (even hole, pyramid, diamond, Kt)-free graphs have bounded treewidth. Our main result is in fact more general, that treewidth is bounded in graphs excluding certain wheels and three-path-configurations, diamonds, and a fixed complete graph. The proof uses "non-crossing decompositions" methods similar to those in previous papers in this series. In previous papers, however, bounded degree was a necessary condition to prove bounded treewidth. The result of this paper is the first to use the method of "non-crossing decompositions" to prove bounded treewidth in a graph class of unbounded maximum degree.

show abstract

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

Cited by 9 publications

References 14 publications

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

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

Submodular functions and perfect graphs

Induced subgraphs and tree decompositions IV. (Even hole, diamond, pyramid)-free graphs

Contact Info

Product

Resources

About