Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Italian domination in generalized Petersen graphs   In a graph $G=(V,E)$, each vertex $v\in V$ is labelled with $0$, $1$ or $2$
such that each vertex labelled with $0$ is adjacent to at least one vertex
labelled $2$ or two vertices labelled $1$. Such kind of labelling is called an
Italian dominating function (IDF) of $G$. The weight of an IDF $f$ is
$w(f)=\sum_{v\in V}f(v)$. The Italian domination number of $G$ is
$\gamma_{I}(G)=\min_{f} w(f)$. Gao et al. (2019) have determined the value of
$\gamma_I(P(n,3))$. In this article, we focus on the study of the Italian
domination number of generalized Petersen graphs $P(n, k)$, $k\neq3$. We
determine the values of $\gamma_I(P(n, 1))$, $\gamma_I(P(n, 2))$ and
$\gamma_I(P(n, k))$ for $k\ge4$, $k\equiv2,3(\bmod5)$ and $n\equiv0(\bmod5)$.
For other $P(n,k)$, we present a bound of $\gamma_I(P(n, k))$. With the
obtained results, we partially solve the open problem presented by Bre\v{s}ar
et al. (2007) by giving $P(n,1)$ is an example for which $\gamma_I=\gamma_{r2}$
and characterizing $P(n,2)$ for which $\gamma_I(P(n,2))=\gamma_{r2}(P(n,2))$.
Moreover, our results imply $P(n,1)$ $(n\equiv0(\bmod\ 4))$ is Italian,
$P(n,1)$ $(n\not\equiv0(\bmod\ 4))$ and $P(n,2)$ are not Italian.

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The categories of this paper are: hep-lat
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The categories of this paper are: nlin
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The categories of this paper are: plasm-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: adap-org
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The categories of this paper are: nucl-th
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The categories of this paper are: chem-ph
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The categories of this paper are: gr-qc
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The categories of this paper are: solv-int
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The categories of this paper are: q-bio
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The categories of this paper are: q-alg
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: dg-ga