Unavoidable vertex-minors in large prime graphs

TL;DR

This paper proves that large prime graphs necessarily contain specific complex structures called vertex-minors, such as cycles or two-clique matchings, highlighting unavoidable configurations in large prime graphs.

Contribution

It establishes the existence of particular vertex-minors in sufficiently large prime graphs, advancing understanding of their structural properties.

Findings

Large prime graphs contain cycles of any fixed length as vertex-minors.

Large prime graphs contain two disjoint cliques of size n joined by a matching as vertex-minors.

There exists a threshold size N beyond which these structures always appear.

Abstract

A graph is prime (with respect to the split decomposition) if its vertex set does not admit a partition (A,B) (called a split) with |A|, |B| >= 2 such that the set of edges joining A and B induces a complete bipartite graph. We prove that for each n, there exists N such that every prime graph on at least N vertices contains a vertex-minor isomorphic to either a cycle of length n or a graph consisting of two disjoint cliques of size n joined by a matching.