Stability of (۱,۲)-total pitchfork domination

Let G=(V, E) be a finite, simple, and undirected graph without isolated vertex. We define a dominating  D of V(G) as a total pitchfork dominating set, if ۱\leq|N(t)\cap V-D|\leq۲ for every t \in D such that G[D] has no isolated vertex. In this paper, the effects of adding or removing an edge and removing a vertex from a graph are studied on the order of minimum total pitchfork dominating set \gamma_{pf}^{t} (G) and the order of minimum inverse total pitchfork dominating set \gamma_{pf}^{-t} (G). Where \gamma_{pf}^{t} (G) is proved here to be increasing by adding an edge and decreasing by removing an edge, which are impossible cases in the ordinary total domination number.

Keywords:

total domination , stability of domination , pitchfork domination