Computational Complexity of Covering Two-vertex Multigraphs with Semi-edges

TL;DR

This paper explores the computational complexity of covering two-vertex multigraphs with semi-edges, providing new classifications and NP-hardness results, extending classical graph covering theory to include semi-edges and their applications.

Contribution

It characterizes the complexity of covering one- and two-vertex multigraphs with semi-edges, including NP-hardness proofs and strengthening previous results for graphs without semi-edges.

Findings

Complete complexity classification for one- and two-vertex graphs with semi-edges.

NP-hardness results for simple input graphs and bipartite target graphs.

Strengthening of known results for graphs without semi-edges.

Abstract

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, Fellows, and Stillwell 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 some solvable cases and, in particular, 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. We remark that our new characterization results also strengthen previously known results for covering graphs without semi-edges, and they in turn apply to an infinite class of simple target graphs with at most two vertices of degree more than two. Some of the results are moreover proven in a more general setting (e.g., finding $k$-tuples of pairwise disjoint perfect matchings in regular graphs).