Homomorphism Reconfiguration via Homotopy.

We consider the following problem for a fixed graph H: given a graph G and two H-colorings of G, i.e. homomorphisms from G to H, can one be transformed into the other by changing one color at a time, maintaining an H-coloring throughout. This is the same as finding a path in the Hom(G, H) complex. For H = K-k this is the problem of finding paths between k-colorings, which was recently shown to be in P for k <= 3 and PSPACE-complete otherwise (Bonsma and Cereceda 2009, Cereceda et al. 2011). We generalize the positive side of this dichotomy by providing an algorithm that solves the problem in polynomial time for any H with no C-4 subgraph. This gives a large class of constraints for which finding solutions to the Constraint Satisfaction Problem is NP-complete, but paths in the solution space can be found in polynomial time. The algorithm uses a characterization of possible reconfiguration sequences (that is, paths in Hom(G, H)), whose main part is a purely topological condition described in terms of the fundamental groupoid of H seen as a topological space.