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 $\sum _{x\in N_G(e)}f(x)\ge 1$ 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 ${\gamma }_{smt}^{\prime }(G)$. Obviously, ${\gamma }_{smt}^{\prime }(G)$ 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.