stability of (1,2)-total pitchfork domination

Let g = (v, e) be a finite, simple, and undirected graph without an isolated vertex. we define a dominating d of v (g) as a total pitchfork dominating set if 1 ≤ |n(t) ∩ v − d| ≤ 2 for every t ∈ 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 γ t pf (g) and the order of minimum inverse total pitchfork dominating set γ −t pf (g). where γ t pf (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.