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 $1leq|n(t)cap v-d|leq2$ 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.