Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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
$\iota'_{k}(G)$, 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 $K_{1,
3}$-free graphs and this result was generalized to $K_{1, r}$-free graphs by
Bollob$\acute{a}$s and Cockayne in 1979. In 2013, Rad and Volkmann proved that
the ratio $i(G)/\gamma(G)$ is at most $\Delta(G)/2$ when $\Delta(G) \in \{3, 4,
5\}$. Further, Furuya et. al. proved that when $\Delta(G) \geq 6$, we have
$i(G)/\gamma(G) \leq \Delta(G) - 2\sqrt{\Delta(G)} + 2$. 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 $\iota'_{k}(G)/\iota_{k}(G) \leq \Delta(G) -
2\sqrt{\Delta(G)} + 2$. Further, we find an inequality between $\iota'_{k}(G)$
and $\iota_{k}(G)$ when $G$ is $K_{1, r}$-free graph. This also generalizes the
result of Bollob$\acute{a}$s and Cockayne.

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The categories of this paper are: nlin
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The categories of this paper are: q-alg
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The categories of this paper are: nucl-th
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The categories of this paper are: physics
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The categories of this paper are: chem-ph
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The categories of this paper are: plasm-ph
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The categories of this paper are: funct-an
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The categories of this paper are: astro-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-ex
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The categories of this paper are: comp-gas
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The categories of this paper are: supr-con
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: atom-ph