Recolouring Homomorphisms to triangle-free reflexive graphs

Abstract

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

Mixing is hard for triangle-free reflexive graphs

“…In [10], we adapted to results of Wrochna [12] for irreflexive graphs to reflexive graphs, and showed that Recon(H) is polynomial time solvable (for reflexive instances) for any triangle-free reflexive graph H. As discussed above, this yields the following corollary. Clearly if H is disconnected then Hom(G, H) is disconnected as soon as G contains an edge, so we always assume that H is connected.…”

“…The proof depends on results from [10] and exploits the same connection to topology that is central to the algorithms there and in [12]. This lends support to the notion that topology may factor into the trichotomy classification.…”

“…This lends support to the notion that topology may factor into the trichotomy classification. We expect the proofs and results of this paper should yield somewhat messier irreflexive versions depending on [12] rather than on [10].…”

“…Though it seems premature to suggest a classification of the graphs H for which these problems fall into the respective complexity classes, some discussion of this is given in [9] for Recon(H). The proofs in [9] and [10] are very topological, and suggest the topology of H may play an important role in the complexity of Recon(H). Indeed, the Homgraph is was originated to apply topological ideas to graph colouring problems, so topological methods are perhaps inevitable.…”

“…In [12], Wrochna, to great effect, extended this idea and showed that for a cycle C mapped to graph H the homotopy type of the image is invariant under reconfiguration. We did the same for reflexive graphs in [10]. In this paper we use the slighly coarser, but computationally simpler topological invariant of homology.…”