Local Connectivity and Cycle Extension 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 \((M(G))\) the induced subgraph of \(G\) on \(M(G)\). We prove that:

1. if \(M(G)\) dominates \(G\) and \(\langle M(G)\rangle \) is connected, then \(G\) is vertex pancyclic orderable,
2. if \(M(G)\) dominates \(G\), \(\langle M(G)\rangle\) is connected, and \(G\setminus M(G)\) is triangle-free, then \(G\) is fully \(2\)-chord extendible,
3. if \(M(G)\) dominates \(G\) and the number of components of \(\langle M(G)\rangle\) does not exceed the connectivity of \(G\), then \(G\) is hamiltonian.