# Graph Homomorphism Reconfiguration and Frozen H-Colourings

For a fixed graph H, the reconfiguration problem for H-colourings (i.e. homomorphisms to H) asks: given a graph G and two H-colourings φ and ψ of G, does there exist a sequence f_0,...,f_m of H-colourings such that f_0=φ, f_m=ψ and f_i(u)f_i+1(v)∈ E(H) for every 0≤ i<m and uv∈ E(G)? If the graph G is loop-free, then this is the equivalent to asking whether it possible to transform φ into ψ by changing the colour of one vertex at a time such that all intermediate mappings are H-colourings. In the affirmative, we say that φ reconfigures to ψ. Currently, the complexity of deciding whether an H-colouring φ reconfigures to an H-colouring ψ is only known when H is a clique, a circular clique, a C_4-free graph, or in a few other cases which are easily derived from these. We show that this problem is PSPACE-complete when H is an odd wheel. An important notion in the study of reconfiguration problems for H-colourings is that of a frozen H-colouring; i.e. an H-colouring φ such that φ does not reconfigure to any H-colouring ψ such that ψ≠φ. We obtain an explicit dichotomy theorem for the problem of deciding whether a given graph G admits a frozen H-colouring. The hardness proof involves a reduction from a CSP problem which is shown to be NP-complete by establishing the non-existence of a certain type of polymorphism.

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