Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: $(1,j)$-set problem in graphs   A subset $D \subseteq V $of a graph $G = (V, E)$ is a $(1, j)$-set if every
vertex $v \in V \setminus D$ is adjacent to at least $1$ but not more than $j$
vertices in D. The cardinality of a minimum $(1, j)$-set of $G$, denoted as
$\gamma_{(1,j)} (G)$, is called the $(1, j)$-domination number of $G$. Given a
graph $G = (V, E)$ and an integer $k$, the decision version of the $(1, j)$-set
problem is to decide whether $G$ has a $(1, j)$-set of cardinality at most $k$.
In this paper, we first obtain an upper bound on $\gamma_{(1,j)} (G)$ using
probabilistic methods, for bounded minimum and maximum degree graphs. Our bound
is constructive, by the randomized algorithm of Moser and Tardos [MT10], We
also show that the $(1, j)$- set problem is NP-complete for chordal graphs.
Finally, we design two algorithms for finding $\gamma_{(1,j)} (G)$ of a tree
and a split graph, for any fixed $j$, which answers an open question posed in
[CHHM13].

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The categories of this paper are: hep-ph
34
The categories of this paper are: chem-ph
28
The categories of this paper are: alg-geom
6
The categories of this paper are: astro-ph
15
The categories of this paper are: nucl-ex
9
The categories of this paper are: nlin
36
The categories of this paper are: cmp-lg
3
The categories of this paper are: physics
25
The categories of this paper are: patt-sol
26
The categories of this paper are: bayes-an
37
The categories of this paper are: supr-con
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
33
The categories of this paper are: plasm-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: gr-qc