Two classes of edge domination in graphs

Let γs′(G) (γl′(G), resp.) be the number of (local) signed edge domination of a graph G   [B. Xu, On signed edge domination numbers of graphs, Discrete Math. 239 (2001) 179–189]. In this paper, we prove mainly that γs′(G)⩽⌊116n-1⌋ and γl′(G)⩽2n-4 hold for any graph G   of order n(n⩾4)n(n⩾4), and pose several open problems and conjectures.