Long dominating cycles in graphs

Let G be a connected graph of order n, and let NC2(G) denote min{|N(u)⌣N(v)|:dist(u,v)=2{, where dist(u,v) is the distance between u and v in G. A cycle C in G is called a dominating cycle, if V(G)/V(C) is an independent set in G. In this paper, we prove that if G contains a dominating cycle and δ ⩾ 2, then G contains a dominating cycle of length at least min{n,2NC2(G) − 3}.