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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Georgia Southern University
2025-05-01
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| Series: | Theory and Applications of Graphs |
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| Online Access: | https://digitalcommons.georgiasouthern.edu/tag/vol12/iss1/6/ |
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| author | Randy R. Davila |
| author_facet | Randy R. Davila |
| author_sort | Randy R. Davila |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-2b1e9da11b9e46cebb48777a8dc85a4d |
| institution | OA Journals |
| issn | 2470-9859 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | Georgia Southern University |
| record_format | Article |
| series | Theory and Applications of Graphs |
| spelling | doaj-art-2b1e9da11b9e46cebb48777a8dc85a4d2025-08-20T02:38:10ZengGeorgia Southern UniversityTheory and Applications of Graphs2470-98592025-05-0112110.20429/tag.2025.120106Lower Bounds for the Total Distance $k$-Domination Number of a GraphRandy R. DavilaFor $k \geq 1$ 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.https://digitalcommons.georgiasouthern.edu/tag/vol12/iss1/6/total distance $k$-dominationtotal domination |
| spellingShingle | Randy R. Davila Lower Bounds for the Total Distance $k$-Domination Number of a Graph Theory and Applications of Graphs total distance $k$-domination total domination |
| title | Lower Bounds for the Total Distance $k$-Domination Number of a Graph |
| title_full | Lower Bounds for the Total Distance $k$-Domination Number of a Graph |
| title_fullStr | Lower Bounds for the Total Distance $k$-Domination Number of a Graph |
| title_full_unstemmed | Lower Bounds for the Total Distance $k$-Domination Number of a Graph |
| title_short | Lower Bounds for the Total Distance $k$-Domination Number of a Graph |
| title_sort | lower bounds for the total distance k domination number of a graph |
| topic | total distance $k$-domination total domination |
| url | https://digitalcommons.georgiasouthern.edu/tag/vol12/iss1/6/ |
| work_keys_str_mv | AT randyrdavila lowerboundsforthetotaldistancekdominationnumberofagraph |