Canonisation and Definability for Graphs of Bounded Rank Width

We prove that the combinatorialWeisfeiler-Leman algorithm of dimension (3k+4) is a complete isomorphism test for the class of all graphs of rank width at most k. Rank width is a graph invariant that, similarly to tree width, measures the width of a certain style of hierarchical decomposition of graphs; it is equivalent to clique width. It was known that isomorphism of graphs of rank width k is decidable in polynomial time (Grohe and Schweitzer, FOCS 2015), but the best previously known algorithm has a running time n(f(k)) for a nonelementary function f. Our result yields an isomorphism test for graphs of rank width k running in time n(O(k)). Another consequence of our result is the first polynomial-time canonisation algorithm for graphs of bounded rank width. Our second main result is that fixed-point logic with counting captures polynomial time on all graph classes of bounded rank width.