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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 (451.78 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28629.1639 | ||
| نویسندگان | ||
| Norah Almalki1؛ Pawaton Kaemawichanurat* 2 | ||
| 1Department of Mathematics and Statistics, College of Science, Taif University | ||
| 2Department of Mathematics, King Mongkut's University of Technology Thonburi | ||
| 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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[8] P. Kaemawichanurat and L. Caccetta, Independence and connectivity of connected domination critical graphs, arXiv preprint arXiv:1911.04961 (2019).
[9] P. Kaemawichanurat, L. Caccetta, and W. Ananchuen, Hamiltonicity of connected domination critical graphs, Ars Combin. 136 (2018), 127–151.
[10] J. Simmons, Closure operations and hamiltonian properties of independent and total domination critical graphs, Ph.D. thesis, University of Victoria, 2005.
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