The Erd\H{o}s-P\'{o}sa property for circle graphs as vertex-minors

TL;DR

This paper establishes a vertex-minor version of the Erdős-Pósa property for circle graphs and extends similar results to binary matroids, linking graph minors, vertex-minors, and matroid theory.

Contribution

It proves a new Erdős-Pósa type property for circle graphs and vertex-minors, and applies these techniques to binary matroids and their minors.

Findings

Existence of a vertex-minor isomorphic to multiple copies of H or a t-perturbation without such minors.

Extension of Erdős-Pósa property to circle graphs and vertex-minors.

Application to binary matroids and their cycle matroid minors.

Abstract

We prove that for any circle graph $H$ with at least one edge and for any positive integer $k$, there exists an integer $t=t(k,H)$ so that every graph $G$ either has a vertex-minor isomorphic to the disjoint union of $k$ copies of $H$, or has a $t$-perturbation with no vertex-minor isomorphic to $H$. Using the same techniques, we also prove that for any planar multigraph $H$, every binary matroid either has a minor isomorphic to the cycle matroid of $kH$, or is a low-rank perturbation of a binary matroid with no minor isomorphic to the cycle matroid of $H$.