On hamiltonicity of P 3-dominated graphs

We introduce a new class of graphs which we call P (3)-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P (3)-dominated graph. We prove that G is hamiltonian if alpha(G (2)) a parts per thousand currency sign kappa(G), with two exceptions: K (2,3) and K (1,1,3). We also prove that G is hamiltonian, if G is 3-connected and |V(G)| a parts per thousand currency sign 5 delta(G) - 5. These results extend known results on (quasi-)claw-free graphs.