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
[1] N. Ananchuen, On domination critical graphs with cut vertices having connected domination number 3, Int. Math. Forum, vol. 2, Citeseer, 2007, pp. 3041–3052. [2] W. Ananchuen, N. Ananchuen, and M.D. Plummer, Vertex criticality for connected domination, Util. Math. 86 (2011), 45--64. [3] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer Publishing Company, Incorporated, 2008. [4] X. Chen, L. Sun, and D. Ma, Connected domination critical graphs, Appl. Math. Lett. 17 (2004), no. 5, 503–507. https://doi.org/10.1016/S0893-9659(04)90118-8 [5] V. Chvátal and P. Erdös, A note on hamiltonian circuits, Discrete Math. 2 (1972), no. 2, 111–113. https://doi.org/10.1016/0012-365X(72)90079-9 [6] P. Kaemawichanurat, On the independence number of 3-cec graphs, Manuscript. [7] P. Kaemawichanurat, Connected domination critical graphs, Ph.D. thesis, Curtin University, 2015. [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. [11] L. Zhang and F. Tian, Independence and connectivity in 3-domination-critical graphs, Discrete Math. 259 (2002), no. 1-3, 227–236. https://doi.org/10.1016/S0012-365X(02)00383-7
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Independence Number and Connectivity of Maximal Connected Domination Vertex Critical Graphs