A simple layered-wheel-like construction

TL;DR

This paper introduces a simple layered-wheel-like graph construction with arbitrarily large treewidth, large girth, and controlled outerstring subgraph treewidth, challenging previous conjectures about graph structure under certain minor exclusions.

Contribution

The authors present a novel, simplified layered-wheel-like graph construction that achieves large treewidth and girth while controlling outerstring subgraph complexity, surpassing previous complex constructions.

Findings

Construction achieves arbitrarily large treewidth and girth.

Outerstring induced subgraphs have bounded treewidth.

Provides counterexamples to a conjecture relating treewidth and induced minors.

Abstract

In recent years, there has been significant interest in characterizing the induced subgraph obstructions to bounded treewidth and pathwidth. While this has recently been resolved for pathwidth, the case of treewidth remains open, and prior work has reduced the problem to understanding the layered-wheel-like obstructions -- graphs that contain large complete minor models with each branching set inducing a path; exclude large walls as induced minors; exclude large complete bipartite graphs as induced minors; and exclude large complete subgraphs. There are various constructions of such graphs, but they are all rather involved. In this paper, we present a simple construction of layered-wheel-like graphs with arbitrarily large treewidth. Three notable features of our construction are: (a) the vertices of degree at least four can be made to be arbitrarily far apart; (b) the girth can be made to be arbitrarily large; and (c) every outerstring induced subgraph of the graphs from our construction has treewidth bounded by an absolute constant. In contrast, among several previously known constructions of layered wheels, none achieves (a); at most one satisfies either (b) or (c); and none satisfies both (b) and (c) simultaneously. In particular, this is related to a former conjecture of Trotignon, that every graph with large enough treewidth, excluding large walls and large complete bipartite graphs as induced minors, and large complete subgraphs, must contain an outerstring induced subgraph of large treewidth. Our construction provides the first counterexample to this conjecture that can also be made to have arbitrarily large girth.