Graphs of bounded depth‐2 rank‐brittleness

Abstract

We characterize classes of graphs closed under taking vertex‐minors and having no Pn and no disjoint union of n copies of the 1‐subdivision of K1,n for some n. Our characterization is described in terms of a tree of radius 2 whose leaves are labeled by the vertices of a graph G, and the width is measured by the maximum possible cut‐rank of a partition of V(G) induced by splitting an internal node of the tree to make two components. The minimum width possible is called the depth‐2 rank‐brittleness of G. We prove that for all n, every graph with sufficiently large depth‐2 rank‐brittleness contains Pn or disjoint union of n copies of the 1‐subdivision of K1,n as a vertex‐minor.