Isomorphism Testing Parameterized by Genus and Beyond

Abstract

We present an isomorphism test for graphs of Euler genus g running in time 2 O(g 4 log g) n O(1) . Our algorithm provides the first explicit upper bound on the dependence on g for an fpt isomorphism test parameterized by the Euler genus of the input graphs. The only previous fpt algorithm runs in time f (g)n for some function f (Kawarabayashi 2015). Actually, our algorithm even works when the input graphs only exclude K 3,h as a minor. For such graphs, no fpt isomorphism test was known before.The algorithm builds on an elegant combination of simple group-theoretic, combinatorial, and graph-theoretic approaches. In particular, we introduce (t, k)-WL-bounded graphs which provide a powerful tool to combine group-theoretic techniques with the standard Weisfeiler-Leman algorithm.