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

arXiv (Cornell University)（2023）

摘要

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 NP-hard problems such as Stable Set, Vertex Cover, Dominating Set and Coloring admit polynomial-time algorithms on this class of graphs.

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关键词

treewidth,subgraphs,decompositions,even-hole-free

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