Long dominating cycles and paths in graphs with large neighborhood unions

Let G be a graph of order n and define NC(G) = min{|N(u) ∪ N(v)| |uv ? E(G)}. A cycle C of G is called a dominating cycle or D-cycle if V(G) - V(C) is an independent set. A D-path is defined analogously. The following result is proved: if G is 2-connected and contains a D-cycle, then G contains a D-cycle of length at least min{n, 2NC(G)} unless G is the Petersen graph. By combining this result with a known sufficient condition for the existence of a D-cycle, a common generalization of Ore's Theorem and several recent “neighborhood union results” is obtained. An analogous result on long D-paths is also established.