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

TL;DR

This paper proves that certain even-hole-free graphs with bounded clique size have logarithmic treewidth, enabling polynomial algorithms for several NP-hard problems within this class.

Contribution

It establishes a logarithmic treewidth bound for a specific class of even-hole-free graphs, confirming a special case of a conjecture and enabling efficient algorithms.

Findings

Treewidth is at most c(t) log n for the class of graphs.

NP-hard problems like Stable Set and Coloring are polynomial-time solvable on these graphs.

The result is tight and used to address the full conjecture in subsequent work.

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.