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The Total Mutual Visibility Number in Bicyclic Graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 25 شهریور 1405 اصل مقاله (457.7 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2026.31659.2919 | ||
| نویسندگان | ||
| Ensieh Golabi Senejani؛ Golam Hossein Fath-Tabar* | ||
| Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan 87317-53153, Iran | ||
| چکیده | ||
| Let $G$ be a graph. A subset of vertices $X \subseteq V(G)$ is termed a total mutual-visibility set if, for any pair of vertices $u, v \in V(G)$, one can find a shortest path connecting them such that none of its internal vertices belong to $X$. The cardinality of the largest such set is known as the total mutual-visibility number, denoted by $\mu_t(G)$. In this study, we determine this number for the class of connected bicyclic graphs. Our analysis categorizes these graphs into three structural types based on the connection between their two cycles: disjoint cycles linked by a path, cycles intersecting at a single vertex, and cycles sharing a common path (Theta-graphs). For each category, we establish explicit formulas to calculate the exact value of $\mu_t(G)$. These results cover all structural configurations, including variations in cycle lengths, leaf attachments, and internal vertex arrangements. Ultimately, this work provides a complete characterization of the total mutual-visibility number for bicyclic graphs and sets a foundation for future research. | ||
| کلیدواژه‌ها | ||
| bicyclic graphs؛ mutual-visibility set؛ total mutual-visibility set؛ total mutual- visibility number | ||
| مراجع | ||
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[1] A. Aljohani and G. Sharma, Complete visibility for mobile robots with lights tolerating faults, International Journal of Networking and Computing 8 (2018), no. 1, 32–52.
[2] S. Bhagat, Optimum algorithm for the mutual visibility problem, Lecture Notes in Computer Science, vol. 12049, Springer, 2020, pp. 31–42. https://doi.org/10.1007/978-3-030-39881-1_4
[3] J.A. Bondy and U.S.R. Murty, Graph Theory, Springer, 2008.
[4] S. Cicerone, A. Di Fonso, G. Di Stefano, and A. Navarra, The geodesic mutual visibility problem for oblivious robots: The case of trees, Proceedings of the 24th International Conference on Distributed Computing and Networking, ACM, 2023, pp. 150–159. [5] S. Cicerone, A. Di Fonso, G. Di Stefano, A. Navarra, and F. Piselli, Mutual and total mutual visibility in hypercube-like graphs, Appl. Math. Comput. 491 (2025), 129216. https://doi.org/10.1016/j.amc.2024.129216
[6] S. Cicerone and G. Di Stefano, Mutual-visibility in distance-hereditary graphs: A linear-time algorithm, Procedia Computer Science 223 (2023), 104–111. https://doi.org/10.1016/j.procs.2023.08.219
[7] S. Cicerone, G. Di Stefano, and S. Klavžar, On the mutual visibility in Cartesian products and triangle-free graphs, Appl. Math. Comput. 438 (2023), 127619. https://doi.org/10.1016/j.amc.2022.127619
[8] G.A. Di Luna, P. Flocchini, S.G. Chaudhuri, F. Poloni, N. Santoro, and G. Viglietta, Mutual visibility by luminous robots without collisions, Inform. and Comput. 254 (2017), 392–418. https://doi.org/10.1016/j.ic.2016.09.005
[9] G. Di Stefano, Mutual visibility in graphs, Appl. Math. Comput. 419 (2022), 126850. https://doi.org/10.1016/j.amc.2021.126850
[10] Z. Hamed-Labbafian, M. Tavakoli, N. Sabeghi, and A. Vesel, Mutual visibility in graphs: hierarchical products, genetic algorithm and some applications, J. Supercomput. 82 (2026), 455. https://doi.org/10.1007/s11227-026-08581-8
[11] Y. Jing, Zero forcing number and general position number in bicyclic graphs, Advances Appl. Math. 12 (2023), no. 4, 1897–1907.
[12] D. Korže and A. Vesel, Mutual-visibility sets in Cartesian products of paths and cycles, Results Math. 79 (2024), no. 3, 116. https://doi.org/10.1007/s00025-024-02139-x
[13] D. Korže and A. Vesel, Variety of mutual-visibility problems in hypercubes, Appl. Math. Comput. 491 (2025), 129218. https://doi.org/10.1016/j.amc.2024.129218 [14] P. Manuel and S. Klavžar, A general position problem in graph theory, Bull. Aust. Math. Soc. 98 (2018), no. 2, 177–187. https://doi.org/10.1017/S0004972718000473
[15] P. Poudel, A. Aljohani, and G. Sharma, Fault-tolerant complete visibility for asynchronous robots with lights under one-axis agreement, Theoret. Comput. Sci. 850 (2021), 116–134. https://doi.org/10.1016/j.tcs.2020.10.033
[16] J. Tian and S. Klavžar, Graphs with total mutual-visibility number zero and total mutual visibility in cartesian products, Discuss. Math. Graph Theory 44 (2024), 1277–1291. | ||
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آمار تعداد مشاهده مقاله: 24 تعداد دریافت فایل اصل مقاله: 40 |
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