Intertwining Connectivities for Vertex-Minors and Pivot-Minors

Abstract

We show that for pairs and of disjoint subsets of vertices of a graph , if is sufficiently large, then there exists a vertex in such that there are two ways to reduce by a vertex-minor operation that removes while preserving the connectivity between and and the connectivity between and . Our theorem implies an analogous theorem of Chen and Whittle (SIAM J. Discrete Math., 28 (2014), pp. 1402--1404) for matroids restricted to binary matroids.