Rank-width and vertex-minors

The rank-width is a graph parameter related in terms of fixed functions to cliquewidth but more tractable. Clique-width has nice algorithmic properties, but no good “minor” relation is known analogous to graph minor embedding for tree-width. In this paper, we discuss the vertex-minor relation of graphs and its connection with rank-width. We prove a relationship between vertex-minors of bipartite graphs and minors of binary matroids, and as an application, we prove that bipartite graphs of sufficiently large rank-width contain certain bipartite graphs as vertex-minors. The main theorem of this paper is that for fixed k, there is a finite list of graphs such that a graph G has rank-width at most k if and only if no graph in the list is isomorphic to a vertex-minor of G. Furthermore, we prove that a graph has rank-width at most 1 if and only if it is distance-hereditary.