  The reconfiguration graph of the $k$-colourings, denoted $\mathcal{R}_k(G)$,
is the graph whose vertices are the $k$-colourings of $G$ and two colourings
are adjacent in $\mathcal{R}_k(G)$ if they differ in colour on exactly one
vertex. In this paper, we investigate the connectivity and diameter of
$\mathcal{R}_{k+1}(G)$ for a $k$-colourable graph $G$ restricted by forbidden
induced subgraphs. We show that $\mathcal{R}_{k+1}(G)$ is connected for every
$k$-colourable $H$-free graph $G$ if and only if $H$ is an induced subgraph of
$P_4$ or $P_3+P_1$. We also start an investigation into this problem for
classes of graphs defined by two forbidden induced subgraphs. We show that if
$G$ is a $k$-colourable ($2K_2$, $C_4$)-free graph, then $\mathcal{R}_{k+1}(G)$
is connected with diameter at most $4n$. Furthermore, we show that
$\mathcal{R}_{k+1}(G)$ is connected for every $k$-colourable ($P_5$,
$C_4$)-free graph $G$.
