At most k-to-1 mappings between graphs, II

Suppose that f is a (⩽ k )-to-1 function from a vertex set of a graph G onto a vertex set of a graph H . We ask when f extends to a continuous (⩽ k )-to-1 map from G onto H . In an earlier paper, the authors answered this question, for the case that k is odd, with local conditions only on the adjacency matrix for H and the inverse adjacency matrix for G . For the present case, when k is even, global conditions are needed and we capture these conditions in a matrix C that exists iff f extends and that describes how the subtle part of the extension must be constructed.