Lower Bounds for the Total Distance $k$-Domination Number of a Graph
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...
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| Main Author: | |
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| Format: | Article |
| Language: | English |
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Georgia Southern University
2025-05-01
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| Series: | Theory and Applications of Graphs |
| Subjects: | |
| Online Access: | https://digitalcommons.georgiasouthern.edu/tag/vol12/iss1/6/ |
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| Summary: | 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 $\gamma_{k}^t(G)$. When $k=1$, the total $k$-domination number reduces to the \emph{total domination number}, written $\gamma_t(G)$; that is, $\gamma_t(G) = \gamma_{1}^t(G)$. This paper shows that several known lower bounds on the total domination number generalize nicely to lower bounds on total distance $k$-domination. |
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| ISSN: | 2470-9859 |