Stability of (1,2)-total pitchfork domination

Document Type : Research Paper


Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq


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 $1\leq|N(t)\cap V-D|\leq2$ 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.