Before revision:  Since a P_t-free graph contains O (n^{t-1}) induced paths in total, branching on such a vertex and recursing independently on the connected components leads to a quasi-polynomial running time bound.
After revision:  Since a P_t-free graph contains \mathcal{O (n^{t-1}) induced paths in total, branching on such a vertex and recursing independently on the connected components leads to a quasi-polynomial running time bound.