On the characterization of trees with signed edge domination numbers 1, 2, 3, or 4

AbstractLet G=(V,E) be a simple graph. For an edge e of G, the closed edge-neighbourhood of e is the set N[e]={e′∈E|e′ is adjacent to e}∪{e}. A function f:E→{1,−1} is called a signed edge domination function (SEDF) of G if ∑e′∈N[e]f(e′)≥1 for every edge e of G. The signed edge domination number of G is defined as γs′(G)=min{∑e∈Ef(e)|f is an SEDF of G}. In this paper, we characterize all trees T with signed edge domination numbers 1, 2, 3, or 4