Secure domination number of k-subdivision of graphs
Nima Ghanbari
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Source: Crossref
Published: Oct 10, 2025
DOI: 10.1142/s1793830925501344
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Let [Formula: see text] be a simple graph. A dominating set of [Formula: see text] is a subset [Formula: see text] such that every vertex not in [Formula: see text] is adjacent to at least one vertex in [Formula: see text]. The cardinality of a smallest dominating set of [Formula: see text], denoted by [Formula: see text], is the domination number of [Formula: see text]. A dominating set [Formula: see text] is called a secure dominating set of [Formula: see text], if for every [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text] is a dominating set of [Formula: see text]. The cardinality of a smallest secure dominating set of [Formula: see text], denoted by [Formula: see text], is the secure domination number of [Formula: see text]. For any [Formula: see text], the [Formula: see text]-subdivision of [Formula: see text] is a simple graph [Formula: see text] which is constructed by replacing each edge of [Formula: see text] with a path of length [Formula: see text]. In this paper, we study the secure domination number of [Formula: see text]-subdivision of [Formula: see text].
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