Abstract
We survey old and recent results on the computational complexity of graph covers, also known as locally bijective graph homomorphisms. This notion opens doors to interesting connections. The motivation itself comes from the classical notion of covering spaces in general topology, graph covers find computer science applications as a model of local computation, and in combinatorics they are used for constructing large highly symmetric graphs.
More than 30 years ago, Abello et al. [1] asked for a complete characterization of the computational complexity of deciding if an input graph covers a fixed one, and until this day only isolated results are known. We look at this question from several different angles of view - covers as locally constrained graph homomorphisms, covers of multigraphs, covers of graphs with semi-edges, or the list variant of the graph covering question. We also mention several open problems, including the Strong Dichotomy Conjecture for graph covers of Bok et al. [6], stating that for every target multigraph H, the H -Cover problem is either polynomial time solvable for arbitrary input graphs, or NP-complete for simple graphs on input. We justify this conjecture for several infinite classes of target (multi)graphs.