We investigate the interplay between the graph isomorphism problem, logical definability, and structural graph theory on a rich family of dense graph classes: graph classes of bounded rank width. 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. A consequence of our result is the first polynomial time canonization algorithm for graphs of bounded rank width.
Our second main result addresses an open problem in descriptive complexity theory: we show that fixed-point logic with counting expresses precisely the polynomial time properties of graphs of bounded rank width.
The question of whether there is an efficient algorithm deciding whether two graphs are isomorphic is one of the oldest and best-known open problems in theoretical computer science. Already mentioned in Karp's18 seminal article on NP-complete combinatorial problems, graph isomorphism (from now on: GI) has remained one of the few natural problems in NP that is neither known to be solvable in polynomial time nor known to be NP-complete. The problem has received considerable attention recently because of Babai's1 breakthrough algorithm deciding GI in quasipolynomial time npolylog(n).
However, the question of whether GI is solvable in polynomial time remains wide open. Polynomial-time algorithms are known for the restrictions of GI to many interesting graph classes, for example the class of planar graphs14, classes of bounded degree19, even all classes excluding a fixed graph as a topological subgraph9, and only recently, graph classes of bounded rank width.11
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