On Signed Edge Domination Numbers of Trees

The signed edge domination number of a graph is an edge variant of the signed domination number. The closed neighbourhood NG[e] of an edge e in a graph G is the set consisting of e and of all edges having a common end vertex with e. Let f be a mapping of the edge set E(G) of G into the set {−1, 1}. If ∑ x∈N [e] f(x) 1 for each e ∈ E(G), then f is called a signed edge dominating function on G. The minimum of the values ∑ x∈E(G) f(x), taken over all signed edge dominating function f on G, is called the signed edge domination number of G and is denoted by γ′ s(G). If instead of the closed neighbourhood NG[e] we use the open neighbourhood NG(e) = NG[e]− {e}, we obtain the definition of the signed edge total domination number γ′ st(G) of G. In this paper these concepts are studied for trees. The number γ′ s(T ) is determined for T being a star of a path or a caterpillar. Moreover, also γ′ s(Cn) for a circuit of length n is determined. For a tree satisfying a certain condition the inequality γ′ s(T ) γ′(T ) is stated. An existence theorem for a tree T with a given number of edges and given signed edge domination number is proved. At the end similar results are obtained for γ′ st(T ).