On hamiltonicity of P <Subscript>3</Subscript>-dominated graphs

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>. We also prove that G is hamiltonian, if G is 3-connected and |V(G)| ≤ 5δ(G) − 5. These results extend known results on (quasi-)claw-free graphs. Copyright Springer-Verlag 2009