Induced subgraphs and tree decompositions V. One neighbor in a hole

TL;DR

This paper investigates the structure of graphs with large treewidth, showing that excluding certain induced subgraphs like wheels and basic obstructions, combined with restrictions on neighbors in holes, bounds the treewidth.

Contribution

It proves that graphs excluding basic obstructions and with limited neighbors in holes have bounded treewidth, extending understanding of graph structure related to treewidth.

Findings

Graphs with large treewidth necessarily contain holes with vertices having at least two neighbors.

Excluding wheels and basic obstructions, along with restrictions on neighbors in holes, bounds the treewidth.

The main theorem establishes bounded treewidth under these conditions.

Abstract

What are the unavoidable induced subgraphs of graphs with large treewidth? It is well-known that the answer must include a complete graph, a complete bipartite graph, all subdivisions of a wall and line graphs of all subdivisions of a wall (we refer to these graphs as the "basic treewidth obstructions"). So it is natural to ask whether graphs excluding the basic treewidth obstructions as induced subgraphs have bounded treewidth. Sintiari and Trotignon answered this question in the negative. Their counterexamples, the so-called "layered wheels," contain wheels, where a wheel consists of a hole (i.e., an induced cycle of length at least four) along with a vertex with at least three neighbors in the hole. This leads one to ask whether graphs excluding wheels and the basic treewidth obstructions as induced subgraphs have bounded treewidth. This also turns out to be false due to Davies' recent example of graphs with large treewidth, no wheels and and no basic treewidth obstructions as induced subgraphs. However, in Davies' example there exist holes and vertices (outside of the hole) with two neighbors in them. Here we prove that a hole with a vertex with at least two neighbors in it is inevitable in graphs with large treewidth and no basic obstruction. Our main result is that graphs in which every vertex has at most one neighbor in every hole (that does not contain it) and with the basic treewidth obstructions excluded as induced subgraphs have bounded treewidth.