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

In the companion paper [Linear rank-width of distance-hereditary graphs I. A polynomial-time 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 $\Phi$ of graphs closed under taking vertex-minors, a graph $G$ is called a vertex-minor obstruction for $\Phi$ if $G\notin \Phi$ but all of its proper vertex-minors are contained in $\Phi$. Secondly, we provide, for each $k\ge 2$, 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 $1$.