On Weak Isomorphism of Rooted Vertex-Colored Graphs

In this work we consider a notion of isomorphism of rooted vertex-colored graphs which allows not only vertices, but also colors to be permuted. Here, a prospective color permutation must be chosen from a group specified at the input. We call this notion weak isomorphism. It turns out that already for severely restricted classes of graphs, the corresponding weak graph isomorphism problem is as hard as the well studied string isomorphism problem. Our main result states that weak isomorphism can be solved in FPT time when simultaneously parameterized by three graph invariants: maximum degree, BFS color number, and BFS width. Intuitively, the second parameter quantifies the number of colors that cross a level of a breadth first search (BFS) tree of the corresponding graph. The third parameter is a width measure based on a BFS-based decomposition introduced independently by Yamazaki et al. [Algorithmica ’99] and by Chepoi and Dragan [Eur. J. Comb. ’00]. We show that the resulting parameterized problem has close relations to the notion of (strong) isomorphism of bounded color class hypergraphs. Our algorithm can be used to solve the latter problem in FPT time. Another consequence is that isomorphism of hypergraphs implicitly represented by ordered decision diagrams (ODD’s) can be solved in FPT time if the width of the involved ODD’s is an additional parameter.