Changing and Unchanging strong efficient edge domination number of some standard graphs when a vertex is removed or an edge is added

Abstract

Let G=(V,\ E) be a simple graph. A subset S of E(G) is a strong (weak) efficient edge dominating set of G if │Ns[e]  S│ = 1 for all e  E(G)(│Nw[e]  S│ = 1 for all e  E(G)) where Ns(e) ={f / f  E(G), f is adjacent to e & deg f ≥ deg e}(Nw(e) ={f / f  E(G), f is adjacent to e & deg f ≤ deg e}) and Ns[e]=Ns(e){e}(Nw[e] = Nw(e){e}). The minimum cardinality of a strong efficient edge dominating set of G (weak efficient edge dominating set of G) is called a strong efficient edge domination number of G and is denoted by {\gamma\prime}_{se}(G) ({\gamma^\prime}_{we}(G)).When a vertex is removed or an edge is added to the graph, the strong efficient edge domination number may or may not be changed. In this paper the change or unchanged of the strong efficient edge domination number of some standard graphs are determined, when a vertex is removed or an edge is added.