A chain theorem for sequentially $3$-rank-connected graphs with respect to vertex-minors

TL;DR

This paper extends the chain theorem to sequentially 3-rank-connected graphs, demonstrating that such graphs have a similar vertex-minor with one fewer vertex, except for small graphs with at most 12 vertices.

Contribution

It introduces a chain theorem for higher connectivity in the context of vertex-minors, generalizing previous results for simple 3-connected graphs and prime graphs.

Findings

Every sequentially 3-rank-connected graph has a similar vertex-minor with one fewer vertex.

The exception is for graphs with 12 or fewer vertices.

The theorem generalizes Tutte's and Bouchet's results to a broader class of graphs.

Abstract

Tutte (1961) proved the chain theorem for simple $3$-connected graphs with respect to minors, which states that every simple $3$-connected graph $G$ has a simple $3$-connected minor with one edge fewer than $G$, unless $G$ is a wheel graph. Bouchet (1987) proved an analog for prime graphs with respect to vertex-minors. We present a chain theorem for higher connectivity with respect to vertex-minors, showing that every sequentially $3$-rank-connected graph $G$ has a sequentially $3$-rank-connected vertex-minor with one vertex fewer than $G$, unless $|V(G)|\leq 12$.