Recolouring Homomorphisms to triangle-free reflexive graphs

TL;DR

This paper proves that recoloring and reconfiguration problems for homomorphisms to triangle-free reflexive graphs can be solved efficiently in polynomial time, expanding understanding of graph homomorphism reconfiguration.

Contribution

It establishes polynomial-time algorithms for recoloring and reconfiguration problems specifically for triangle-free reflexive graphs, a previously unresolved case.

Findings

Recoloring problem is polynomial-time solvable for these graphs.

Reconfiguration problem can be decided efficiently in the same setting.

Results extend the class of graphs with known efficient reconfiguration algorithms.

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 $H$.