Edge \(k\)-to-\(1\) Homomorphisms

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 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.