Binary matroids and degree-boundedness for pivot-minors

TL;DR

This paper establishes bounded average degree for certain bipartite graphs excluding specific pivot-minors, utilizing binary matroid theory, and provides tight bounds for degrees in bipartite circle graphs.

Contribution

It introduces a new boundedness result for classes of bipartite graphs defined by pivot-minor exclusions, based on the binary matroid structure theorem.

Findings

Bounded average degree for $K_{s,s}$-subgraph-free graphs excluding a fixed pivot-minor.

Vertex degree bounds in bipartite circle graphs are tight and depend on parameters s and t.

Application of the binary matroid structure theorem to graph minor problems.

Abstract

We prove that for every bipartite graph $H$ and positive integer $s$, the class of $K_{s,s}$-subgraph-free graphs excluding $H$ as a pivot-minor has bounded average degree. Our proof relies on the announced binary matroid structure theorem of Geelen, Gerards, and Whittle. Along the way, we also prove that every $K_{s,t}$-free bipartite circle graph with $s\le t$ has a vertex of degree at most $\max\{2s-2, t-1\}$ and provide examples showing that this is tight.