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

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