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arXiv · 2506.03973

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

Rutger Campbell·J. Pascal Gollin·Meike Hatzel·O-joung Kwon·Rose McCarty·Sang-il Oum·Sebastian Wiederrecht
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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 $H$.

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Rutger Campbell, J. Pascal Gollin, Meike Hatzel, O-joung Kwon, Rose McCarty, Sang-il Oum, Sebastian Wiederrecht. 2025-06-04. The Erd\H{o}s-P\'{o}sa property for circle graphs as vertex-minors. https://arxiv.org/abs/2506.03973

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