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

In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomialtime algorithm, Algorithmica 78(1):342–377, 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 Φ of graphs closed under taking vertex-minors, a graph G is called a vertex-minor obstruction for Φ if G R Φ but all of its proper vertex-minors are contained in Φ. Secondly, we provide, for each k ě 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.