The grid theorem for vertex-minors

TL;DR

This paper proves that large rank-width graphs necessarily contain any fixed circle graph as a vertex-minor, establishing a fundamental link between graph complexity and the presence of specific substructures.

Contribution

It establishes a universal property of graphs with large rank-width, showing they always contain any circle graph as a vertex-minor, extending the understanding of graph minors.

Findings

Large rank-width graphs contain all circle graphs as vertex-minors.

The result generalizes previous theorems relating graph width parameters to minors.

Provides a new structural insight into the relationship between rank-width and vertex-minors.

Abstract

We prove that, for each circle graph $H$, every graph with sufficiently large rank-width contains a vertex-minor isomorphic to $H$.