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^{-۱} of V-D is an inverse triple effect dominating set if every v \in D^{-۱} dominates exactly three vertices of V-D^{-۱}. The inverse triple effect domination number \gamma_{t e}^{-۱}(G) is the minimum cardinality over all inverse triple effect dominating sets in G. Some results and properties on \gamma_{t e}^{-۱}(G) are given and proved. Under any conditions the graph satisfies \gamma_{t e}(G)+\gamma_{t e}^{-۱}(G)=n is studied. Lower and upper bounds for the size of a graph that has \gamma_{t e}^{-۱}(G) are putted in two cases when D^{-۱}=V-D and when D^{-۱} \neq V-D . Which properties of a vertex to be belongs to D^{-۱} or out of it are discussed. Then, \gamma_{t e}^{-۱}(G) is evaluated and proved for several graphs.

کلیدواژه ها:

dominating set ، Triple effect domination ، Inverse triple effect domination