Computational Complexity of Covering Disconnected Multigraphs

The notion of graph covers is a discretization covering spaces introduced and deeply studied in topology. In discrete mathematics theoretical computer science, they have attained lot attention from both the structural complexity perspectives. Nonetheless, disconnected graphs were usually omitted considerations with explanation that it sufficient to understand coverings connected components target by source one. However, different (but equivalent) versions definition generalize nonequivalent definitions graphs. aim this paper summarize issue compare three approaches graphs: 1) locally bijective homomorphisms, 2) globally surjective homomorphisms (which we call covers), 3) which cover every vertex same number times equitable covers). standpoint our comparison deciding if an input fixed graph. We show satisfy what certainly natural welcome property: polynomial time decidable such for component graph, NP-complete at least one its components. Despite this, argue third variant, covers, right one, when considering colored (multi)graphs. Moreover, differ parameter point view. conclude complete characterization 2-vertex multigraphs semi-edges. present results utmost generality strength. accord current trends consider (multi)graphs semi-edges, and, on other hand, proving NP-completeness simple