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

TL;DR

This paper proves that graphs excluding certain configurations, including three-path-configurations and triangles, have logarithmic treewidth, enabling polynomial algorithms for several NP-hard problems.

Contribution

It generalizes previous results by establishing logarithmic treewidth bounds for broader classes of graphs excluding specific configurations.

Findings

Proves (theta, triangle)-free graphs have treewidth at most c log |V(G)|

Shows excluding three-path-configurations leads to logarithmic treewidth

Enables polynomial algorithms for NP-hard problems in these graph classes

Abstract

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 Coloring admit polynomial time algorithms in graphs excluding the three-path-configurations and a fixed complete graph.