We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em graphs with semi-edges}. The notion of graph covering is a discretization of coverings between surfaces or topological spaces, a notion well known and deeply studied in classical topology. Graph covers have found applications in discrete mathematics for constructing highly symmetric graphs, and in computer science in the theory of local computations. In 1991, Abello et al. asked for a classification of the computational complexity of deciding if an input graph covers a fixed target graph, in the ordinary setting (of graphs with only edges). Although many general results are known, the full classification is still open. In spite of that, we propose to study the more general case of covering graphs composed of normal edges (including multiedges and loops) and so-called semi-edges. Semi-edges are becoming increasingly popular in modern topological graph theory, as well as in mathematical physics. They also naturally occur in the local computation setting, since they are lifted to matchings in the covering graph. We show that the presence of semi-edges makes the covering problem considerably harder; e.g., it is no longer sufficient to specify the vertex mapping induced by the covering, but one necessarily has to deal with the edge mapping as well. We show some solvable cases, and completely characterize the complexity of the already very nontrivial problem of covering one- and two-vertex (multi)graphs with semi-edges. Our NP-hardness results are proven for simple input graphs, and in the case of regular two-vertex target graphs, even for bipartite ones. This provides a strengthening of previously known results for covering graphs without semi-edges, and may contribute to better understanding of this notion and its complexity.

中文翻译：

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半边覆盖两个顶点多重图的计算复杂度

我们开始研究{\ em带半边图}的图覆盖率（又称局部双射图同态）的计算复杂性。图形覆盖的概念是表面或拓扑空间之间的覆盖的离散化，这是在经典拓扑结构中众所周知并经过深入研究的概念。图覆盖已发现在离散数学中用于构建高度对称图的应用，以及在计算机科学中的局部计算理论中的应用。1991年，Abello等人。在普通设置（仅具有边的图）中，要求对确定输入图是否覆盖固定目标图的计算复杂度进行分类。尽管已知许多常规结果，但仍可以进行完整分类。尽管如此，我们建议研究由法线边（包括多边和环）和所谓的半边组成的覆盖图的更一般情况。半边沿在现代拓扑图论以及数学物理学中正变得越来越流行。它们也自然出现在本地计算设置中，因为它们被提升到覆盖图中的匹配项。我们证明了半边的存在使覆盖问题变得更加困难。例如，仅仅指定由覆盖物引起的顶点映射不再是足够的，但是必须也必须处理边缘映射。我们展示了一些可解决的情况，并完全刻画了用半边覆盖一个和两个顶点（多）图这一本来就很不容易的问题的复杂性。我们的NP硬度结果已通过简单的输入图证明，对于常规的两顶点目标图，甚至对于二部图也是如此。这为覆盖不带半边的图形提供了先前已知结果的增强，并且可能有助于更好地理解此概念及其复杂性。