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A study on strong and geodetic domination integrity sets in graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 05 فروردین 1404 اصل مقاله (2.05 M) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2025.30077.2300 | ||
| نویسندگان | ||
| Zehui Shao1؛ Saeed Kosari* 2؛ Sundareswaran Raman3؛ Balaraman Ganesan4 | ||
| 1Institute of Computing Science and Technology, Guangzhou University, Guangzhou, 510006, China | ||
| 2Huangpu Research School of Guangzhou University, Guangzhou, China | ||
| 3Department of Mathematics, Sri Sivasubramaniya Nadar College of Engineering, OMR, Chennai, Tamilnadu, India | ||
| 4Department of Mathematics, St. Joseph’s Institute of Technology, OMR, Chennai, Tamilnadu, India | ||
| چکیده | ||
| Consider a graph $\Omega = (\mathcal{V,E})$ that is simple, and let $\vartheta_1$ and $\vartheta_2$ be elements of $\mathcal{V}(\Omega)$ suth that $\vartheta_1 \vartheta_2 \in \mathcal{E}(\Omega)$. Then, $\vartheta_1$ is said to strongly dominate $\vartheta_2$ if $deg\left(\vartheta_1\right) \geq deg \left(\vartheta_2\right)$. A set $K$ of $\mathcal{V}(\Omega)$ is identified as a strong dominating set ($sd$-set) if every vertex $\vartheta_2$ outside of $K$ is strongly dominated by at least one node $\vartheta_1$ within $K$. The concept of strong domination integrity for $\Omega$ is defined as $\widetilde{SDI}(\Omega) = \mathop{min}_{K \subseteq \mathcal{V}}\{|K| + m(\Omega - K): K$ is a $sd$-set of $\Omega$\}. Similarly, the set $K \subseteq \mathcal{V} (\Omega)$ is identified as a geodetic dominating set ($gd$-set) if $K$ is both geodetic and dominating set. The geodetic domination integrity of $\Omega$ is defined as $\widetilde{GDI} (\Omega) = min \{|K| + m(\Omega - K): K$ is a $gd$-set of $\Omega\}$. This paper delves into the study of strong and geodetic domination integrity sets, as well as the impact of node removal on these sets. Additionally, it introduces the concepts of $\widetilde{SDI}$-Excellent and $\widetilde{GDI}$-Excellent graphs, provides examples, and derives theorems from these graphs. | ||
| کلیدواژهها | ||
| Strong domination integrity sets؛ Geodetic domination integrity sets؛ $\widetilde{SDI}$-Ext graphs؛ $\widetilde{GDI}$-Ext graphs | ||
| مراجع | ||
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