Characterizations of the graphs with dominating parameters
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912603791425536 |
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| author | Ma, Yuhan |
| author_facet | Ma, Yuhan |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10170 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterizations of the graphs with dominating parameters Ma, Yuhan Combinatorics 05C38, 05C35, 05C40 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. |
| title | Characterizations of the graphs with dominating parameters |
| topic | Combinatorics 05C38, 05C35, 05C40 |
| url | https://arxiv.org/abs/2410.10170 |