Local connectivity and cycle xtension in claw-free graphs

Let G be a connected claw-free graph, M (G) the set of all vertices of G that have a connected neighborhood, and hM (G)i the induced subgraph of G on M (G). We prove that (i) if M (G) dominates G and hM (G)i is connected, then G is vertex pan-cyclic orderable, (ii) if M (G) dominates G; hM (G)i is connected, and G n M (G) is triangle-free, then G is fully 2-chord extendable, (iii) if M (G) dominates G and the number of components of hM (G)i does not exceed the connectivity of G, then G is hamiltonian.