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Total domination versus triad domination | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 16 آبان 1404 اصل مقاله (421.8 K) | ||
| نوع مقاله: Special issue of CCO to honor Odile Favaron | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.30952.2676 | ||
| نویسندگان | ||
| Teresa W. Haynes* 1؛ Michael A Henning2 | ||
| 1Department of Mathematics and Statistics, East Tennessee State University, Johnson City, TN 37614-0002 USA | ||
| 2Department of Mathematics and Applied Mathematics, University of Johannesburg, Auckland Park, 2006 South Africa | ||
| چکیده | ||
| A dominating set in a graph $G$ is a set $S$ of vertices of $G$ such that every vertex in $V(G) \setminus S$ is adjacent to a vertex in $S$. A total dominating set in $G$ is a dominating set $S$ with the additional property that the subgraph $G[S]$ induced by $S$ is isolate-free. A triad dominating set $S$ (also called a $3$-component dominating set in the literature) is a dominating set in which every component in $G[S]$ has order at least~$3$. The triad domination number, denoted $\gamma_{td}(G)$, of $G$ is the minimum cardinality among all triad dominating sets of $G$. We observe that $\gamma(G) \le \gamma_t(G) \le \gamma_{td}(G)$, where $\gamma(G)$ is the domination number of $G$ and $\gamma_t(G)$ is the total domination number of $G$. We show that the ratio $\frac{\gamma_{td}(G)}{\gamma_t(G)}$ is at most $\frac{3}{2}$. We establish properties of the graphs $G$ satisfying $\gamma_{td}(G) = \frac{3}{2}\gamma_t(G)$ and characterize the trees achieving this equality. | ||
| کلیدواژهها | ||
| Domination؛ total domination؛ triad domination؛ $3$-component domination | ||
| مراجع | ||
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[11] W. Yang and B. Wu, Dominating sets inducing large component in graphs with minimum degree two, Graphs Combin. 39 (2023), no. 5, Article number: 99. https://doi.org/10.1007/s00373-023-02687-z
[12] W. Yang and B. Wu, Proof of a conjecture on dominating sets inducing large component in graphs with minimum degree two, Discrete Math. 347 (2024), no. 10, 114122. https://doi.org/10.1016/j.disc.2024.114122 | ||
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