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∈D−1 dominates exactly three vertices of V−D−1. The inverse triple effect domination number γ−1te(G) is the minimum cardinality over all inverse triple effect dominating sets in G. Some results and properties on γ−1te(G) are given and proved. Under any conditions the graph satisfies γte(G)+γ−1te(G)=n is studied. Lower and upper bounds for the size of a graph that has γ−1te(G) are putted in two cases when D−1=V−D and when D−1≠V−D. Which properties of a vertex to be belongs to D−1 or out of it are discussed. Then, γ−1te(G) is evaluated and proved for several graphs.