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

Let 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 γ s ′ ( G ) = min { ∑ e ∈ E f ( 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.