Tree-width dichotomy

TL;DR

This paper characterizes when the tree-width of graphs in certain hereditary classes is bounded, based on the presence of specific forbidden induced subgraphs, providing a precise dichotomy.

Contribution

It establishes a complete characterization of hereditary graph classes with bounded tree-width using a finite set of forbidden induced subgraphs.

Findings

Bounded tree-width occurs iff the forbidden set includes specific graphs.

Identifies key forbidden subgraphs that determine tree-width bounds.

Provides a dichotomy criterion for hereditary graph classes.

Abstract

We prove that the tree-width of graphs in a hereditary class defined by a finite set $F$ of forbidden induced subgraphs is bounded if and only if $F$ includes a complete graph, a complete bipartite graph, a tripod (a forest in which every connected component has at most 3 leaves) and the line graph of a tripod.