Canonisation and Definability for Graphs of Bounded Rank Width

Abstract

We prove that the combinatorial Weisfeiler-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 non-elementary 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. Preliminaries GraphsA graph is a pair G = (V, E) with vertex set

Graph Isomorphism for $$(H_1,H_2)$$-Free Graphs: An Almost Complete Dichotomy

“…Theorem 2 ([20]). For every constant c, Graph Isomorphism is polynomial-time solvable on graphs of clique-width at most c.Grohe and Neuen [17] have since improved this result by showing that the more general Canonisation problem is also in XP when parameterized by clique-width. 5 A graph is (H1, H2)-free if it has no induced subgraph isomorphic to H1 or H2.…”

The graph isomorphism problem

“…There are also natural graph classes that do not exclude a fixed graph as a minor (and that are therefore not covered by Theorem 3.5). The WL dimension of the class of interval graphs is 2 [27] and the WL dimension of the class of graphs of rank width at most k is at most 3k + 4 [37]. Contrasting all these results, Theorem 3.4 shows that the WL dimension of the class of 3-regular graphs is ∞, even though isomorphism of 3-regular graphs can be decided in polynomial time [58] (cf.…”

Separating Rank Logic from Polynomial Time

“…These include e.g. graphs with excluded minors [16] and graphs with bounded rank width [17]. While showing that rank logic defines the CFI query for prime fields is rather simple, for CPT this is a non-trivial result.…”