Unavoidable induced subgraphs in graphs with complete bipartite induced minors

TL;DR

This paper establishes that graphs containing large complete bipartite induced minors necessarily include small cycles or theta subgraphs, revealing limitations in bounding treewidth through forbidden induced minors.

Contribution

It proves new induced minor conditions that guarantee the presence of specific subgraphs, advancing understanding of graph structure related to induced minors and treewidth.

Findings

Graphs with $K_{134, 12}$ as an induced minor contain small cycles or thetas.

Graphs with $K_{3, 4}$ as an induced minor contain triangles or thetas.

Excluding certain induced minors does not bound treewidth or tree-independence number.

Abstract

We prove that if a graph contains the complete bipartite graph $K_{134, 12}$ as an induced minor, then it contains a cycle of length at most~12 or a theta as an induced subgraph. With a longer and more technical proof, we prove that if a graph contains $K_{3, 4}$ as an induced minor, then it contains a triangle or a theta as an induced subgraph. Here, a \emph{theta} is a graph made of three internally vertex-disjoint chordless paths $P_1 = a \dots b$, $P_2 = a \dots b$, $P_3 = a \dots b$, each of length at least two, such that no edges exist between the paths except the three edges incident to $a$ and the three edges incident to $b$. A consequence is that excluding a grid and a complete bipartite graph as induced minors is not enough to guarantee a bounded tree-independence number, or even that the treewidth is bounded by a function of the size of the maximum clique, because the existence of graphs with large treewidth that contain no triangles or thetas as induced subgraphs is already known (the so-called layered wheels).