Lower bounds for the total (distance) $k$-domination number of a graph
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929384892399616 |
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| author | Davila, Randy |
| author_facet | Davila, Randy |
| contents | For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total (distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than itself. The \emph{total (distance) $k$-domination number} of $G$ is the minimum cardinality of a total $k$-dominating set in $G$, and is denoted by $γ_{k}^t(G)$. When $k=1$, the total $k$-domination number reduces to the \emph{total domination number}, written $γ_t(G)$; that is, $γ_t(G) = γ_{1}^t(G)$. This paper shows that several known lower bounds on the total domination number generalize nicely to lower bounds on total (distance) $k$-domination. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_08770 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Lower bounds for the total (distance) $k$-domination number of a graph Davila, Randy Combinatorics For $k \geq 1$ and a graph $G$ without isolated vertices, a \emph{total (distance) $k$-dominating set} of $G$ is a set of vertices $S \subseteq V(G)$ such that every vertex in $G$ is within distance $k$ to some vertex of $S$ other than itself. The \emph{total (distance) $k$-domination number} of $G$ is the minimum cardinality of a total $k$-dominating set in $G$, and is denoted by $γ_{k}^t(G)$. When $k=1$, the total $k$-domination number reduces to the \emph{total domination number}, written $γ_t(G)$; that is, $γ_t(G) = γ_{1}^t(G)$. This paper shows that several known lower bounds on the total domination number generalize nicely to lower bounds on total (distance) $k$-domination. |
| title | Lower bounds for the total (distance) $k$-domination number of a graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2406.08770 |