Algorithmic Aspects of Regular Graph Covers

A graph G covers a graph H if there exists a locally bijective homomorphism from G to H. We deal with regular covers where this homomorphism is prescribed by the action of a semiregular subgroup of Aut(G). We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for given graphs G and H whether G regularly covers H. When |H| = 1, this problem becomes Cayley graph recognition for which the complexity is still unresolved. Another special case arises for |G| = |H| when it becomes the graph isomorphism problem. Our main result is an involved FPT algorithm solving RegularCover for planar inputs G in time O * (2 e(H)/2) where e(H) denotes the number of edges of H. The algorithm is based on dynamic programming and employs theoretical results proved in a related structural paper. Further, when G is 3-connected, H is 2-connected or the ratio |G|/|H| is an odd integer, we can solve the problem RegularCover in polynomial time. In comparison, Bílka et al. (2011) proved that testing general graph covers is NP-complete for planar inputs G when H is a small fixed graph such as K 4 or K 5. 1. Introduction. The notion of covering originates in topology as a notion of local similarity of two topological spaces. For instance, consider the unit circle and the real line. Globally, these two spaces are not the same, they have different properties, different fundamental groups, etc. But when we restrict ourselves to a small part of the circle, it looks the same as a small part of the real line; more precisely the two spaces are locally homeomorphic, and thus they share the local properties. The notion of covering formalizes this property of two spaces being locally the same. Suppose that we have two topological spaces: a big one G and a small one H. We say that G covers H if there exists an epimorphism called a covering projection p : G → H which locally preserves the structure of G. For instance, the mapping p(x) = (cos x, sin x) from the real line to the unit circle is a covering projection. The existence of a covering projection ensures that G looks locally the same as H; see Fig. 1.1a. In this paper, we study coverings of graphs in a more restricting version called regular covering, for which …