Intertwining connectivities for vertex-minors and pivot-minors

TL;DR

This paper proves that large graphs contain a vertex whose removal via vertex-minor operations preserves specific connectivities, extending a matroid theorem to graph theory.

Contribution

It establishes a new theorem linking vertex-minors and connectivities in graphs, generalizing a prior matroid result to graph structures.

Findings

Existence of a special vertex in large graphs preserving connectivities

Extension of Chen and Whittle's matroid theorem to graphs

Implications for vertex-minor operations and graph connectivity

Abstract

We show that for pairs $(Q,R)$ and $(S,T)$ of disjoint subsets of vertices of a graph $G$, if $G$ is sufficiently large, then there exists a vertex $v$ in $V(G)-(Q\cup R\cup S\cup T)$ such that there are two ways to reduce $G$ by a vertex-minor operation that removes $v$ while preserving the connectivity between $Q$ and $R$ and the connectivity between $S$ and $T$. Our theorem implies an analogous theorem of Chen and Whittle (2014) for matroids restricted to binary matroids.