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.

An inverse triple effect domination in graphs Keywords:

dominating set , Triple effect domination , Inverse triple effect domination