Graphs of bounded depth-2 rank-brittleness

Abstract

We characterize classes of graphs closed under taking vertex-minors and having noPnand no disjoint union ofncopies of the 1-subdivision ofK1,nfor somen. Our characterization is described in terms of a tree of radius 2 whose leaves are labeled by the vertices of a graphG, and the width is measured by the maximum possible cut-rank of a partition ofV(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 ofG. We prove that for alln, every graph with sufficiently large depth-2 rank-brittleness containsPnor disjoint union ofncopies of the 1-subdivision ofK1,nas a vertex-minor.