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

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.