Algorithmic Aspects of Regular Graph Covers

TL;DR

This paper investigates the computational complexity of regular graph covers, introducing an FPT algorithm for planar graphs and polynomial-time solutions for specific connectivity cases, advancing understanding of graph covering problems.

Contribution

It presents the first fixed-parameter tractable algorithm for regular covers of planar graphs and identifies polynomial-time solvable cases based on connectivity and ratio conditions.

Findings

FPT algorithm for regular covers of planar graphs with time $O^*(2^{e(H)/2})$

Polynomial-time algorithms for 3-connected $G$ or 2-connected $H$ cases

Regular cover testing is NP-complete in general for planar graphs with small fixed $H$

Abstract

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 $\textrm{Aut}(G)$. 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, B\'ilka 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 $K_5$.