Signed Edge Majority Total Domination Numbers in Graphs

Abstract

We initiate the study of signed edge majority total domination in graphs. The open neighborhood \(N_G(e)\) of an edge \(e\) in a graph \(G\) is the set consisting of all edges having a common 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 at least half of the edges \(e \in E(G)\), then \(f\) is called a signed edge majority total dominating function of \(G\). The value \(\sum_{e\in E(G)}f(e)\), taking the minimum over all signed edge majority total dominating functions \(f\) of \(G\), is called the signed edge majority total domination number of \(G\) and denoted by \(\gamma’_{smt}(G)\). Obviously, \(\gamma’_{smt}(G)\) is defined only for graphs \(G\) which have no connected components isomorphic to \(K_2\). In this paper, we establish lower bounds on the signed edge majority total domination number of forests.