🤖 AI Summary
This paper investigates the computational complexity of locally bijective covering problems for graphs with semi-edges, focusing on deciding whether an input graph covers a given target multigraph—allowing loops, multiple edges, and semi-edges—with one or two vertices. We provide the first systematic complexity classification of this generalized covering problem: (i) we establish that semi-edges introduce intrinsic hardness, rendering edge-mapping non-negligible; (ii) we fully classify the P vs. NP-complete dichotomy for all single- and two-vertex target graphs; (iii) we prove that covering is NP-complete for almost all two-vertex target graphs with semi-edges—even when restricted to regular or bipartite targets; and (iv) we precisely characterize all polynomial-time solvable cases. Our approach integrates combinatorial graph theory, modeling via local bijective homomorphisms, and carefully constructed NP-hardness reductions tailored to semi-edge structures.
📝 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 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.
