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 $\phi$ and $\psi$ of $G$, does there exist a sequence $f_0,\dots ,f_m$ of $H$-colourings such that $f_0=\phi$, $f_m=\psi$ and $f_i(u)f_{i+1}(v)\in E(H)$forevery$0\leq i<m$and$uv\in E(G)$?Ifthegraph$G$ is loop-free, then this is the equivalent to asking whether it possible to transform $\phi$ into $\psi$ by changing the colour of one vertex at a time such that all intermediate mappings are $H$-colourings. In the affirmative, we say that $\phi$ reconfigures to $\psi$. Currently, the complexity of deciding whether an $H$-colouring $\phi$ reconfigures to an $H$-colouring $\psi$ 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 $\phi$ such that $\phi$ does not reconfigure to any $H$-colouring $\psi$ such that $\psi \mathrm{\ne }\phi$. 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.