Covering Regular Graphs

A covering projection from a graphGonto a graphHis a “local isomorphism”: a mapping from the vertex set ofGonto the vertex set ofHsuch that, for everyv∈V(G), the neighborhood ofvis mapped bijectively onto the neighborhood (inH) of the image ofv. We investigate two concepts that concern graph covers of regular graphs. The first one is called “multicovers”: we show that for any regular graphHthere exists a graphGthat allows many different covering projections ontoH. Secondly, we considerpartial covers, which require only thatGbe a subgraph of a cover ofH. As an application of our results we show that there are infinitely many rigid regular graphsHfor which theH-cover problem—deciding if a given graphGcoversH—is NP-complete. This resolves an open problem related to the characterization of graphsHfor whichH-COVER is tractable.