Reconfiguring homomorphisms to reflexive graphs via a simple reduction

초록

Given a graph G and two graph homomorphisms alpha and /B from G to a fixed graph H, the problem H-recoloring asks whether there is a transformation from alpha to /B that changes the image of a single vertex at each step and keeps a graph homomorphism throughout. The complexity of the problem depends, among other things, on the presence of loops on the vertices. We provide a simple reduction that, using a known algorithmic result for H-recoloring for square-free irreflexive graphs H, yields a polynomial-time algorithm for H-recoloring for square-free reflexive graphs H. This generalizes all known algorithmic results for H-recoloring for reflexive graphs H. Furthermore, the construction allows us to reprove some of the known hardness results. Finally, we provide a partial inverse of the construction for bipartite instances. (c) 2025 Published by Elsevier Ltd.