Reconfiguring homomorphisms to reflexive graphs via a simple reduction

Given a graph $G$ and two graph homomorphisms $\alpha$ and $\beta$ from $G$ to a fixed graph $H$, the problem $H$-Recoloring asks whether there is a transformation from $\alpha$ to $\beta$ 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 recover some of the known hardness results. Finally, we provide a partial inverse of the construction for bipartite instances.