Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions

In the companion paper (Adler et al., 2017), we presented a characterization of the linear rank-width of distance-hereditary graphs, from which we derived an algorithm to compute it in polynomial time. In this paper, we investigate structural properties of distance hereditary graphs based on this characterization. First, we prove that for a fixed tree T, every distance-hereditary graph of sufficiently large linear rank-width contains a vertex minor isomorphic to T. We extend this property to bigger graph classes, namely, classes of graphs whose prime induced subgraphs have bounded linear rank-width. Here, prime graphs are graphs containing no splits. We conjecture that for every tree T, every graph of sufficiently large linear rank-width contains a vertex minor isomorphic to T. Our result implies that it is sufficient to prove this conjecture for prime graphs. For a class Phi of graphs closed under taking vertex-minors, a graph G is called a vertex-minor obstruction for Phi if G is not an element of Phi, but all of its proper vertex-minors are contained in Phi. Secondly, we provide, for each k >= 2, a set of distance-hereditary graphs that contains all distance-hereditary vertex-minor obstructions for graphs of linear rank-width at most k. Also, we give a simpler way to obtain the known vertex-minor obstructions for graphs of linear rank-width at most 1. (C) 2018 Elsevier Ltd. All rights reserved.