Recoloring via modular decomposition

The reconfiguration graph of the $k$-colorings of a graph $G$, denoted $R{k}(G)$, is the graph whose vertices are the $k$-colorings of $G$ and two colorings are adjacent in $R{k}(G)$ if they differ in color on exactly one vertex. A graph $G$ is said to be recolorable if $R{\ell}(G)$ is connected for all $\ell \geq \chi(G)$+1. We use the modular decomposition of several graph classes to prove that the graphs in the class are recolorable. In particular, we prove that every ($P5$, diamond)-free graph, every ($P5$, house, bull)-free graph, and every ($P5$, $C_5$, co-fork)-free graph is recolorable.