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On the semitotal dominating sets of graphs | ||
| Journal of Algebra and Related Topics | ||
| دوره 12، شماره 2، فروردین 2025، صفحه 37-46 اصل مقاله (298.79 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2024.23672.1496 | ||
| نویسندگان | ||
| H. Zaherifar؛ S. Alikhani* | ||
| Department of Mathematical Sciences, Yazd University, Yazd, Iran | ||
| چکیده | ||
| A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$. The semitotal domination number of $G$ is the minimum cardinality of a semitotal dominating set of $G$ and is denoted by $\gamma_{t2}(G)$. In this paper after computation of semitotal domination number of specific graphs, we count the number of this kind of dominating sets of arbitrary size in some graphs. | ||
| کلیدواژهها | ||
| Dominating set؛ Semitotal domination number؛ Product | ||
| مراجع | ||
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1. S. Akbari, S. Alikhani and Y.H. Peng, Characterization of graphs using domination polynomial, Europ. J. Combin., 31 (2010), 1714-1724. 2. S. Alikhani, M.H. Akhbari, C. Eslahchi, R. Hasni, On the number of outer connected dominating sets of graphs, Utilitas Math. 91 (2013), 99-107. 3. S. Alikhani, S., Jahari and M. Mehryar, Counting the number of weakly connected dominating sets of graphs, Malaysian J. Math. Sci. (3) 10 (2016), 299-308. 4. S. Alikhani, J.I. Brown and S. Jahari, On the domination polynomials of friendship graphs, Filomat, (1) 30 (2016), 169-178. 5. S. Alikhani and Y.H. Peng, Introduction to domination polynomial of a graph, Ars Combin. 114 (2014), 257-266. 6. D. Bauer, F. Harary, J. Nieminen and C. Su el, Domination alternation sets in graphs, Discrete Math., 47 (1983), 153-161. 7. W. Goddard, M.A. Henning, and C.A. McPillan, Semitotal domination in graphs, Util. Math. 94 (2014), 67-81. 8. T. W. Haynes, S.T. Hedetniemi and P. J. Slater, Fundamentals of domination in graphs, Marcel Dekker, NewYork, 1998 . 9. M.A. Henning and A. Yeo, Total Domination in Graphs, Springer, New York, 2013. 10. M.A. Henning and A.J. Marcon, On matching and semitotal domination in graphs, Discrete Math. 324 (2014), 13-18. 11. M.A. Henning and A.J. Marcon, Vertices contained in all or in no minimum semitotal dominating set of a tree, Discuss. Math. Graph Theory, 36 (2016), 71-93. 12. M.A. Henning, S. Pal and D. Pradhan, The semitotal domination problem in block graphs, Discuss. Math. Graph Theory, 42 (2022) 231-248. | ||
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