Inequalities between Partial Domination and Independent Partial Domination in Graphs

For a graph $G$, a vertex subset $S\subseteq V(G)$ is said to be $K_k$-isolating if $G-N_G[S]$ does not contain $K_k$ as a subgraph. The $K_k$-isolation number of $G$, denoted by ${\iota }_k(G)$, is the minimum cardinality of a $K_k$-isolating set of $G$. Analogously, $S$ is said to be independent $K_k$-isolating if $S$ is a $K_k$-isolating set of $G$ and $G[S]$ has no edge. The independent $K_k$-isolation number of $G$, denoted by ${\iota }_k^{\mathrm{\prime }}(G)$, is the minimum cardinality of an independent $K_k$-isolating set of $G$. Clearly, when $k=1$, we have $\gamma (G)={\iota }_1(G)$ and $i(G) = \iota’{1}(G)$where$\gamma(G)$and$i(G)$ are the domination and independent domination numbers. For classic results between $\gamma (G)$ and $i(G)$, in 1978, Allan and Laskar proved that $\gamma (G)=i(G)$ for all $K{1, 3}$-freegraphsandthisresultwasgeneralizedto$K_{1, r}$-free graphs by Bollob$\acute{a}$s and Cockayne in 1979. In 2013, Rad and Volkmann proved that the ratio $i(G)\mathrm{/}\gamma (G)$ is at most $\mathrm{\Delta }(G)\mathrm{/}2$ when $\Delta(G) \in {3, 4, 5}$\mathrm{.}Further,Furuyaet\mathrm{.}al\mathrm{.}provedthatwhen$\Delta(G) \geq 6$,wehave$i(G)/\gamma(G) \leq \Delta(G) - 2\sqrt{\Delta(G)} + 2$. In this paper, for a smallest $K_k$-isolating set $S$, we prove that $\iota’k(G)\le -\frac{\iota_k^2(G)}{\ell} +i_k(G)(\Delta +2)-\ell \Delta$where$\ell$ is the number of some specific vertices of $S$ such that the union of their closed neighborhoods in $S$ is $S$. We prove that this bound is sharp. A special case of our main theorem implies $\iota’{k}(G)/\iota_{k}(G) \leq \Delta(G) - 2\sqrt{\Delta(G)} + 2$\mathrm{.}Further,wefindaninequalitybetween$\iota’{k}(G)$and$\iota{k}(G)$when$G$is$K_{1, r}$-free graph. This also generalizes the result of Bollob$\acute{a}$s and Cockayne.