Determining a graph from its reconfiguration graph

Given a graph $G$ and a natural number $k$, the $k$-recolouring graph $\mathcal{C}_k(G)$ is the graph whose vertices are the $k$-colourings of $G$ and whose edges link pairs of colourings which differ at exactly one vertex of $G$. Recently, Hogan et al. proved that $G$ can be determined from $\mathcal{C}_k(G)$ provided $k$ is large enough (quadratic in the number of vertices of $G$). We improve this bound by showing that $k=χ(G)+1$ colours suffice, and provide examples of families of graphs for which $k=χ(G)$ colours do not suffice. We then extend this result to $k$-Kempe-recolouring graphs, whose vertices are again the $k$-colourings of a graph $G$ and whose edges link pairs of colourings which differ by swapping the two colours in a connected component induced by selecting those two colours. We show that $k=χ(G)+2$ colours suffice to determine $G$ in this case. Finally, we investigate the case of independent set reconfiguration, proving that in only a few trivial cases is one guaranteed to be able to determine a graph $G$.