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arXiv · 2410.10170

Characterizations of the graphs with dominating parameters

Yuhan Ma
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Abstract

A subset $S$ of vertices of $G$ is a \textit{dominating set} of $G$ if every vertex in $V(G)-S$ has a neighbor in $S$. The \textit{domination number} \(γ(G)\) is the minimum cardinality of a dominating set of $G$. A dominating set $S$ is a \textit{total dominating set} if $N(S)$=$V$ where $N(S)$ is the neighbor of $S$. The \textit{total domination number} \(γ_t(G)\) equals the minimum cardinality of a total dominating set of $G$. A set $D$ is an \textit{isolate set} if the induced subgragh $G[D]$ has at least one isolated vertex. The \textit{isolate number} \(i_0(G)\) is the minimum cardinality of a maximal isolate set. In this paper we study these parameters and answer open problems proposed by Hamid et al. in 2016.

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Yuhan Ma. 2025-09-25. Characterizations of the graphs with dominating parameters. https://arxiv.org/abs/2410.10170

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