Pitchfork Domination and It's Inverse for Corona and Join Operations in Graphs

Abstract

Let $G$ be a finite simple and undirected graph without isolated vertices. A subset $D$ of $V$ is a pitchfork dominating set if every vertex $v \in D$ dominates at least $j$ and at most $k$ vertices of $V-D$, where $j$ and $k$ are non-negative integers .The domination number of $G$, denoted by $\gamma_{pf}(G)$ is a minimum cardinality over all pitchfork dominating sets in $G$. A subset $D^{-1}$ of $V-D$ is an inverse pitchfork dominating set if $D^{-1}$ is a pitchfork dominating set. The inverse domination number of $G$, denoted by $\gamma_{pf}^{-1}(G)$ is a minimum cardinality over all inverse pitchfork dominating sets in $G$. In this paper, the pitchfork domination and the inverse pitchfork domination are determined when $j=1$ and $k=2$ for some graphs that obtained from graph operations corona and join.