Independent partial domination

For $p\in(0,1]$, a set $S\subseteq V$ is said to $p$-dominate or partially dominate a graph $G = (V, E)$ if $\frac{|N[S]|}{|V|}\geq p$. The minimum cardinality among all $p$-dominating sets is called the $p$-domination number and it is denoted by $\gamma_{p}(G)$. Analogously, the independent partial domination ($i_p(G)$) is introduced and studied here independently and in relation with the classical domination. Further, the partial independent set and the partial independence number $\beta_p(G)$ are defined and some of their properties are presented. Finally, the partial domination chain is established as $\gamma_p(G)\leq i_p(G)\leq \beta_p(G) \leq \Gamma_p(G)$.