Complexity of partial covers of graphs

Abstract

A graph G partially covers a graph H if it allows a locally injective homomorphism from G to H, i.e. an edge-preserving vertex mapping which is injective on the closed neighborhood of each vertex of G. The notion of partial covers is closely related to the generalized frequency assignment problem. We study the computational complexity of the question whether an input graph G partially covers a fixed graph H. Since this problem is at least as difficult as deciding the existence of a full covering projection (a locally bijective homomorphism), we concentrate on classes of problems (described by parameter graphs H) for which the full cover problem is polynomially solvable. In particular, we treat graphs H which contain at most two vertices of degree greater than two, and for such graphs we exhibit both NP-complete and polynomially solvable instances. The techniques are based on newly introduced notions of generalized matchings and edge precoloring extension of bipartite graphs. © 2001 Springer Berlin Heidelberg.