Stability of (1,2)-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 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 γtpf(G) and the order of minimum inverse total pitchfork dominating set γ−tpf(G). Where γtpf(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.