Lower bounds on signed edge total domination numbers in graphs

Abstract

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 $$ \sum\limits_{x \in N_G (e)} {f(x) \geqslant 1} $$ for each e ∈ E(G), then f is called a signed edge total dominating function of G. The minimum of the values $$ \sum\limits_{e \in E(G)} {f(e)} $$ , 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 γ st ′(G). Obviously, γ 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 γ st ′(G). In particular, we prove that γ st ′(T) ⩾ 2 − m/3 for every tree T of size m ⩾ 2. We also classify all trees T with γ st ′(T).