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 α(G 2 ) ≤ κ(G), with two exceptions: K 2,3 and K...

We introduce a new class of graphs which we call P <Subscript>3</Subscript>-dominated graphs. This class properly contains all quasi-claw-free graphs, and hence all claw-free graphs. Let G be a 2-connected P <Subscript>3</Subscript>-dominated graph. We prove that G is hamiltonian if α(G <Superscript>2</Superscript>) ≤ κ(G), with two exceptions: K <Subscript>2,3</Subscript> and K <Subscript>1,1,3</Subscript>....</subscript></subscript></superscript></subscript></subscript>