Edge degrees and dominating cycles

The edge degree d(e) of the edge e=uv is defined as the number of neighbours of e, i.e., |N(u)∪N(v)|-2. Two edges are called remote if they are disjoint and there is no edge joining them. In this article, we prove that in a 2-connected graph G, if d(e₁)+d(e₂)>|V(G)|-4 for any remote edges e₁,e₂, then all longest cycles C in G are dominating, i.e., G-V(C) is edgeless. This lower bound is best possible.As a corollary, it holds that if G is a 2-connected triangle-free graph with σ₂(G)>|V(G)|/2, then all longest cycles are dominating.