Intertwining Connectivities for Vertex-Minors and Pivot-Minors

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