Lower bounds on signed edge total domination numbers in graphs

The open neighborhood NG(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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum\limits_{x \in N_G (e)} {f(x) \geqslant 1} $$\end{document} for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \sum\limits_{e \in E(G)} {f(e)} $$\end{document}, 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 γst′(G). Obviously, γst′(G) is defined only for graphs G which have no connected components isomorphic to K2. In this paper we present some lower bounds for γst′(G). In particular, we prove that γst′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γst′(T).