The open neighborhood $N_G(e)$ of an edge $e$ in a graph $G$ is the set consisting of all edges having a common end-vertex with $e$. Let $f$ be a function on $E(G)$, the edge set of $G$, into the set $\{-1, 1\}$. If $ \sum _{x\in N_G(e)}f(x) \geq 1$ for each $e\in E(G)$, then $f$ is called a signed edge total dominating function of $G$. The minimum of the values $\sum _{e\in E(G)} f(e)$, taken over all signed edge total dominating function $f$ of $G$, is called the signed edge total domination number of $G$ and is denoted by $\gamma _{st}'(G)$. Obviously, $\gamma _{st}'(G)$ is defined only for graphs $G$ which have no connected components isomorphic to $K_2$. In this paper we present some lower bounds for $\gamma _{st}'(G)$. In particular, we prove that $\gamma _{st}'(T)\geq 2-m/3$ for every tree $T$ of size $m\geq 2$. We also classify alltrees $T$ with $\gamma_{st}'(T)=2-m/3$.
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).