A graph $G$ is \emph{$P_5$-reducible} if every vertex of $G$ lies in at most one induced $P_5$ (path on five vertices). We show that a number of interesting results concerning $P_5$-free graphs can be extended to $P_5$-reducible graphs, namely: the existence of a dominating clique or $P_3$, the fact that $k$-colorability can be decided in polynomial time (for fixed $k$), and the fact that a maximum stable set can be found in polynomial time in the class of $k$-colorable $P_5$-reducible graphs (for fixed $k$).