Recolouring Homomorphisms to triangle-free reflexive graphs

Abstract

For a graph $H$, the $H$-recolouring problem $\operatorname{Recol}(H)$ 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 $\operatorname{Recon}(H)$ of deciding whether two given homomorphisms from a given graph $G$ to $H$ are in the same component of the Hom-graph $\operatorname{{Hom}}(G,H)$, can be solved in polynomial time for triangle-free reflexive graphs $H$.