The length of dominating cycle of claw-free graph

The length of dominating cycles is usually discussed in control problems. A dominating cycle of a graph G is a cycle C of G such that V(G) - V(C) is an independent set. In this article, we prove that for any claw-free graph G with delta(G) >= 2, the length of longest dominating cycle is at least minfn, n,2 vertical bar NC(2()c)(G)vertical bar I}, where NC2(G) denotes the vertex set N(u) [ (N)(v) containing the minimum number of vertices for all vertices u, v with d(u, v)= 2 in G.