Induced subgraphs and tree decompositions V. Small components of big vertices

TL;DR

This paper investigates the structure of graphs with large treewidth and small high-degree components, establishing conditions under which such graphs contain large subdivided walls or their line graphs, extending understanding of obstacles to bounded treewidth.

Contribution

It proves that graphs with small high-degree components and large treewidth necessarily contain large subdivided walls or their line graphs, refining the characterization of complex graph structures.

Findings

Graphs with small high-degree components and large treewidth contain large subdivided walls.

The main result is optimal, as it fails for larger degree thresholds, confirmed by Davies' example.

Provides insight into structures causing large treewidth beyond usual suspects.

Abstract

Aboulker, Adler, Kim, Sintiari, and Trotignon conjectured that every graph with bounded maximum degree and large treewidth must contain, as an induced subgraph, a large subdivided wall, or the line graph of a large subdivided wall. This conjecture was recently proved by Korhonen, but the problem of identifying the obstacles to bounded treewidth in the general case (that is, without the bounded maximum degree condition) remains wide open. Examples of structures of large treewidth which avoid the "usual suspects" have been constructed by Sintiari and Trotignon, and by Davies. In this note, we aim to better isolate the features of these examples that lead to large treewidth. To this end, we prove the following result. Let $G$ be a graph, and write $\gamma(G)$ for the size of a largest connected component in the graph induced by $G$ on the set of vertices of degree at least 3. If $\gamma(G)$ is small and the treewidth of $G$ is large, then $G$ must contain a large subdivided wall or the line graph of a large subdivided wall. This result is the best possible, in the sense that the conclusion fails if we replace 3 by any larger number in the definition of $\gamma(G)$, as evidenced by Davies' example.