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.