Linear time algorithms or hamiltonian problems on (claw, net)-free graphs

We prove that claw-free graphs, containing an induced dominating path, have a Hamiltonian path, and that 2-connected claw-free graphs, containing an i nduced doubly dominating cycle or a pair of vertices such that there exist two internally disjoint induced dominating paths connecting them, have a Ha miltonian cycle. As a consequence, we obtain linear time algorithms for bot h problems if the input is restricted to (claw, net)-free graphs. These gra phs enjoy those interesting structural properties.