In line with the recent development in topological graph theory, we are considering undirected graphs that are allowed to contain {\em multiple edges}, {\em loops}, and {\em semi-edges}. A graph is called {\em simple} if it contains no semi-edges, no loops, and no multiple edges. A graph covering projection, also known as a locally bijective homomorphism, is a mapping between vertices and edges of two graphs which preserves incidences and which is a local bijection on the edge-neighborhood of every vertex. This notion stems from topological graph theory, but has also found applications in combinatorics and theoretical computer science. It has been known that for every fixed simple regular graph $H$ of valency greater than 2, deciding if an input graph covers $H$ is NP-complete. Graphs with semi-edges have been considered in this context only recently and only partial results on the complexity of covering such graphs are known so far. In this paper we consider the list version of the problem, called \textsc{List-$H$-Cover}, where the vertices and edges of the input graph come with lists of admissible targets. Our main result reads that the \textsc{List-$H$-Cover} problem is NP-complete for every regular graph $H$ of valency greater than 2 which contains at least one semi-simple vertex (i.e., a vertex which is incident with no loops, with no multiple edges and with at most one semi-edge). Using this result we show the NP-co/polytime dichotomy for the computational complexity of \textsc{ List-$H$-Cover} for cubic graphs.

翻译：根据拓扑图论的最新发展，我们考虑允许包含{\em 多重边}、{\em 环}和{\em 半边}的无向图。若图不含半边、环和多重边，则称为{\em 简单图}。图覆盖投影（亦称局部双射同态）是两图顶点与边之间的映射，该映射保持关联关系，且每个顶点的边邻域上为局部双射。这一概念源于拓扑图论，在组合数学与理论计算机科学中也有应用。已知对于每个度数大于2的固定简单正则图$H$，判定输入图是否覆盖$H$是NP完全的。仅近期才在此背景下考虑含半边的图，且目前仅知此类图覆盖问题的部分复杂性结果。本文考虑该问题的列表版本，称为\textsc{List-$H$-Cover}，其中输入图的顶点和边附有可容许目标的列表。我们的主要结论表明：对于每个度数大于2且至少包含一个半简单顶点（即：不与环关联、无多重边、且至多与一条半边关联的顶点）的正则图$H$，\textsc{List-$H$-Cover}问题是NP完全的。基于此结果，我们给出了三次图\textsc{List-$H$-Cover}计算复杂性的NP完全/多项式时间二分法。