On Signed Edge Total Domination Numbers of Graphs

For the terminology and notations not defined here, we adopt those in Bondy and Murty [1] and Xu [2] and consider simple graphs only. Let G = (V,E) be a graph with vertex set V = V (G) and edge set E = E(G). For any vertex v ∈ V , NG(v) denotes the open neighborhood of v in G and NG[v] = NG(v) ∪ {v} the closed one. dG(v) = |NG(v)| is called the degree of v in G, ∆ and δ denote the maximum degree and minimum degree of G, respectively. Similarly, if e = uv ∈ E, NG(e) denotes the open edgeneighborhood of e in G and NG[e] = NG(e) ∪ {e} the closed one. dG(e) is called the degree of e in G, ∆e and δe denote the maximum edge degree and minimum edge degree of G, respectively. If the graph is clear from the context, NG(v), NG[v], dG(v) and NG(e), NG[e], dG(e) can simply be denoted by N(v), N [v], d(v) and N(e), N [e], d(e). If d(v) is odd (even), then v is called an odd (even) vertex of G. Similarly, if d(e) is even (odd), then e is called an even (odd) edge and d(e) = d(u) + d(v)− 2. In this paper, we define Eo = {e ∈ E|d(e)is odd } and Ee = {e ∈ E|d(e) is even}. In recent years, several kinds of domination problems in graphs have been investigated [2–6]. Most of them belong to the vertex domination of graphs, such as signed domination [7, 8], minus domination [8], majority domination, etc. Recently, the problem has been changed from vertex domination to edge domination, and a few results have been obtained about the edge domination