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arXiv · 2307.13684

Induced subgraphs and tree decompositions X. Towards logarithmic treewidth for even-hole-free graphs

Tara Abrishami·Bogdan Alecu·Maria Chudnovsky·Sepehr Hajebi·Sophie Spirkl
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Abstract

A generalized $t$-pyramid is a graph obtained from a certain kind of tree (a subdivided star or a subdivided cubic caterpillar) and the line graph of a subdivided cubic caterpillar by identifying simplicial vertices. We prove that for every integer $t$ there exists a constant $c(t)$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ and no induced subgraph isomorphic to a generalized $t$-pyramid has treewidth at most $c(t)\log{n}$. This settles a special case of a conjecture of Sintiari and Trotignon; this bound is also best possible for the class. It follows that several \textsf{NP}-hard problems such as \textsc{Stable Set}, \textsc{Vertex Cover}, \textsc{Dominating Set} and \textsc{Coloring} admit polynomial-time algorithms on this class of graphs. Results from this paper are also used in later papers of the series, in particular to solve the full version of the Sintiari-Trotignon conjecture.

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Tara Abrishami, Bogdan Alecu, Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl. 2023-07-25. Induced subgraphs and tree decompositions X. Towards logarithmic treewidth for even-hole-free graphs. https://arxiv.org/abs/2307.13684

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