Independence Number and Connectivity of Maximal Connected Domination Vertex Critical Graphs

Almalki, Norah; Kaemawichanurat, Pawaton

doi:10.22049/cco.2023.28629.1639

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	Independence Number and Connectivity of Maximal Connected Domination Vertex Critical Graphs
Communications in Combinatorics and Optimization
مقاله 1، دوره 9، شماره 2، شهریور 2024، صفحه 185-196 اصل مقاله (451.77 K)
نوع مقاله: Original paper
شناسه دیجیتال (DOI): 10.22049/cco.2023.28629.1639
نویسندگان
Norah Almalki¹؛ Pawaton Kaemawichanurat^* ²^{، 3}
¹Department of Mathematics and Statistics, College of Science, Taif University, Saudi Arabia
²Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology, Thonburi, Thailand
³Mathematics and Statistics with Applications (MaSA), Bangkok, Thailand
چکیده
A $k$-CEC graph is a graph $G$ which has connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. A $k$-CVC graph $G$ is a $2$-connected graph with $\gamma_{c}(G) = k$ and $\gamma_{c}(G - v) < k$ for any $v \in V(G)$. A graph is said to be maximal $k$-CVC if it is both $k$-CEC and $k$-CVC. Let $\delta$, $\kappa$, and $\alpha$ be the minimum degree, connectivity, and independence number of $G$, respectively. In this work, we prove that for a maximal $3$-CVC graph, if $\alpha = \kappa$, then $\kappa = \delta$. We additionally consider the class of maximal $3$-CVC graphs with $\alpha < \kappa$ and $\kappa < \delta$, and prove that every $3$-connected maximal $3$-CVC graph when $\kappa < \delta$ is Hamiltonian connected.
کلیدواژه‌ها
connected domination؛ independence number؛ connectivity

مراجع
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Independence Number and Connectivity of Maximal Connected Domination Vertex Critical Graphs