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 Φ of graphs closed under taking vertex-minors, a graph G
is called a vertex-minor obstruction for Φ if G∈/Φ but all of
its proper vertex-minors are contained in Φ. 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.Comment: 38 pages, 13 figures, 1 table, revised journal version. A preliminary
version of Section 5 appeared in the proceedings of WG1