An inverse triple effect domination in graphs

In this paper, an inverse triple effect domination is introduced for any finite graph $G=(V, E)$ simple and undirected without isolated vertices. A subset $D^{-1}$ of $V-D$ is an inverse triple effect dominating set if every $v in D^{-1}$ dominates exactly three vertices of $V-D^{-1}$. The inverse triple effect domination number $gamma_{t e}^{-1}(G)$ is the minimum cardinality over all inverse triple effect dominating sets in $G$. Some results and properties on $gamma_{t e}^{-1}(G)$ are given and proved. Under any conditions the graph satisfies $gamma_{t e}(G)+gamma_{t e}^{-1}(G)=n$ is studied. Lower and upper bounds for the size of a graph that has $gamma_{t e}^{-1}(G)$ are putted in two cases when $D^{-1}=V-D$ and when $D^{-1} neq V-D .$ Which properties of a vertex to be belongs to $D^{-1}$ or out of it are discussed. Then, $gamma_{t e}^{-1}(G)$ is evaluated and proved for several graphs.