Induced subgraphs and tree decompositions XVI. Complete bipartite induced minors

TL;DR

This paper establishes a structural dichotomy for graphs with large complete bipartite induced minors, showing they must contain either a large wall or a large constellation, and characterizes the unavoidable subgraphs within these constellations.

Contribution

It introduces a new structural theorem linking large bipartite induced minors to specific subgraph configurations, advancing understanding of graph minors.

Findings

Graphs with large bipartite induced minors contain large walls or constellations.

Characterization of unavoidable subgraphs in large constellations.

Refinement of the structure of constellations in graph minors.

Abstract

We prove that for every graph $G$ with a sufficiently large complete bipartite induced minor, either $G$ has an induced minor isomorphic to a large wall, or $G$ contains a large constellation; that is, a complete bipartite induced minor model such that on one side of the bipartition, each branch set is a singleton, and on the other side, each branch set induces a path. We further refine this theorem by characterizing the unavoidable induced subgraphs of large constellations as two types of highly structured constellations. These results will be key ingredients in several forthcoming papers of this series.