Recolouring homomorphisms to triangle-free reflexive graphs

Abstract

For a graph H, the H-recolouring problem \({\text {Recol}}(H)\) asks, for two given homomorphisms from a given graph G to H, if one can get between them by a sequence of homomorphisms of G to H in which consecutive homomorphisms differ on only one vertex. We show that, if G and H are reflexive and H is triangle-free, then this problem can be solved in polynomial time. This shows, at the same time, that the closely related H-reconfiguration problem \({\text {Recon}}(H)\) of deciding whether two given homomorphisms from a given graph G to H are in the same component of the Hom-graph \({\text {{Hom}}}(G,H)\), can be solved in polynomial time for triangle-free reflexive graphs H.