Computational Complexity of Covering Twovertex Multigraphs with Semiedges
Abstract
We initiate the study of computational complexity of graph coverings, aka locally bijective graph homomorphisms, for {\em graphs with semiedges}. 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 socalled semiedges. Semiedges 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 semiedges 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 twovertex (multi)graphs with semiedges. Our NPhardness results are proven for simple input graphs, and in the case of regular twovertex target graphs, even for bipartite ones. This provides a strengthening of previously known results for covering graphs without semiedges, and may contribute to better understanding of this notion and its complexity.
 Publication:

arXiv eprints
 Pub Date:
 March 2021
 arXiv:
 arXiv:2103.15214
 Bibcode:
 2021arXiv210315214B
 Keywords:

 Computer Science  Discrete Mathematics;
 Computer Science  Computational Complexity;
 Mathematics  Combinatorics