Algorithmic Aspects of Regular Graph Covers with Applications to Planar Graphs

A graph $G$ covers a graph $H$ if there exists a locally bijective homomorphism from $G$ to $H$. We deal with regular covers in which this locally bijective homomorphism is prescribed by an action of a subgroup of ${\rm Aut}(G)$. Regular covers have many applications in constructions and studies of big objects all over mathematics and computer science. We study computational aspects of regular covers that have not been addressed before. The decision problem RegularCover asks for two 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. Therefore, we restrict ourselves to graph classes with polynomially solvable graph isomorphism. Inspired by Negami, we apply the structural results used by Babai in the 1970's to study automorphism groups of graphs. Our main result is the following FPT meta-algorithm: Let $\cal C$ be a class of graphs such that the structure of automorphism groups of 3-connected graphs in $\cal C$ is simple. Then we can solve RegularCover for $\cal C$-inputs $G$ in time $O^*(2^{e(H)/2})$ where $e(H)$ denotes the number of the edges of $H$. As one example of $\cal C$, this meta-algorithm applies to planar graphs. In comparison, testing general graph covers is known to be NP-complete for planar inputs $G$ even for small fixed graphs $H$ such as $K_4$ or $K_5$. Most of our results also apply to general graphs, in particular the complete structural understanding of regular covers for 2-cuts.