Hamiltonian cycles and 2-dominating induced cycles in claw-free graphs

Let G = (V, E) be a connected graph. For a vertex subset S subset of V, G[S] is the subgraph of G induced by S. A cycle C (a path, respectively) is said to be an induced cycle (path, respectively) if G[V(C)] = C (G[V(P)] = P, respectively). The distance between a vertex x and a subgraph H of G is denoted by d(x, H) = min{d(x, y) | y is an element of V(H)}, where d(x, y) is the distance between x and y. A subgraph H of G is called 2-dominating if d(x, H) <= 2 for all x is an element of V(G). An induced path P of G is said to be maximal if there is no induced path P' satisfying V(P) subset of V(P') and V(P') \ V(P) not equal empty set. In this paper, we assume that G is a connected claw-free graph satisfying the following condition: for every maximal induced path P of length p >= 2 with end vertices u, v it holds: d(u) + d(v) >= |V(G)| - p + 2. Under this assumption, we prove that G has a 2-dominating induced cycle and G is Hamiltonian.