Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Algorithmic Complexity of Isolate Secure Domination in Graphs   A dominating set $S$ is an Isolate Dominating Set (IDS) if the induced
subgraph $G[S]$ has at least one isolated vertex. In this paper, we initiate
the study of new domination parameter called, isolate secure domination. An
isolate dominating set $S\subseteq V$ is an isolate secure dominating set
(ISDS), if for each vertex $u \in V \setminus S$, there exists a neighboring
vertex $v$ of $u$ in $S$ such that $(S \setminus \{v\}) \cup \{u\}$ is an IDS
of $G$. The minimum cardinality of an ISDS of $G$ is called as an isolate
secure domination number, and is denoted by $\gamma_{0s}(G)$. Given a graph $
G=(V,E)$ and a positive integer $ k,$ the ISDM problem is to check whether $ G
$ has an isolate secure dominating set of size at most $ k.$ We prove that ISDM
is NP-complete even when restricted to bipartite graphs and split graphs. We
also show that ISDM can be solved in linear time for graphs of bounded
tree-width.

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The categories of this paper are: chem-ph
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The categories of this paper are: comp-gas
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The categories of this paper are: q-alg
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The categories of this paper are: mtrl-th
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: solv-int
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The categories of this paper are: funct-an
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The categories of this paper are: ao-sci
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The categories of this paper are: econ
3
The categories of this paper are: physics
4
The categories of this paper are: cond-mat
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The categories of this paper are: math
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The categories of this paper are: quant-ph
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The categories of this paper are: gr-qc