Complexity of Graph Covering Problems

For a xed graph H, the H-cover problem asks whether an input graph G allows a degree preserving mapping f : V (G) ! V (H) such that for every v 2 V (G), f(N G (v)) = N H (f(v)). In this paper we design eecient algorithms for certain graph covering problems according to two basic techniques. The rst is based in part on a reduction to the 2-SAT problem. The second technique exploits necessary and suucient conditions for the partition of a graph into 1-factors and 2-factors. For other innnite classes of graph covering problems we derive NP-completeness results by reductions from graph coloring problems. We illustrate this methodology by classifying the complexity of all H-cover problems deened by simple graphs H with at most 6 vertices. 1 Motivation and overview For a xed graph H, the H-cover problem admits a graph G as input and asks about the existence of a \local isomorphism": a labeling of vertices of G by vertices of H so that for every vertex v 2 V (G) the multiset of labels of its neighbors is equal to the neighborhood (in H) of the label of v. We trace this concept to Conway and Biggs' construction of innnite classes of highly symmetric graphs, see Chapter 19 of 3]. Graph coverings are special cases of covering spaces from algebraic topology 15] and have many applications in topological graph theory 6]. In an algorithmic framework, graph coverings have been applied by Angluin to study \local knowledge" in distributed computing environments 2] and by Courcelle and M etivier 4] to show that nontrivial minor closed classes of graphs cannot be recognized by local computations. Abello et al. 1] raised the question of computational complexity of H-cover problems, noting that there are both polynomial-time solvable (easy) and NP-complete (diicult) versions of this problem for diierent graphs H. In 10] we initiated a general study of the computational complexity of H-cover problems. In a later paper 11] 1 we made signiicant headway on the conjecture that H-cover problems are NP-complete for k-regular graphs whenever k 3. A very recent paper investigates the complexity of the graph covering problem for colored graphs 12]. In this paper, we extend and complete the methodology introduced in 10] to analyze the complexity of certain graph covering problems. The paper is organized as follows. First, we introduce our vocabulary by giving the necessary …