Edge \(k\)-to-1 homomorphisms (Q2715942)

A homomorphism from a graph to another graph is an edge preserving vertex mapping. A homomorphism naturally induces an edge mapping of the two graphs. If, for each edge in the image graph, its preimages have \(k\) elements, then we have an edge \(k\)-to-1 homomorphism. We characterize the connected graphs which admit an edge 2-to-1 homomorphism to a path, or to a cycle. A special case of edge \(k\)-to-1 homomorphism---\(k\)-wrapped quasicovering---is also considered.