List Covering of Regular Multigraphs

A graph covering projection, also known as a locally bijective homomorphism, is mapping between vertices and edges of two graphs which preserves incidencies local bijection. This notion stems from topological theory, but has found applications in combinatorics theoretical computer science. It been that for every fixed simple regular H valency greater than 2, deciding if an input covers NP-complete. In recent years, theory developed into heavily relying on multiple edges, loops, semi-edges, only partial results the complexity multigraphs with semi-edges are so far. this paper we consider list version problem, called List- $$H\text {-}\textsc {Cover}$$ , where come lists admissible targets. Our main result reads problem NP-complete multigraph 2 contains at least one semi-simple vertex (i.e., incident no most semi-edge). Using almost show NP-co/polytime dichotomy computational cubic multigraphs, leaving just five open cases.