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Vatandoost, ‎Ebrahim, Abbasi, ‎Ahmad, Mirasheh, Kamran. (1405). Proof of a conjecture for the watching number. فناوری آموزش, 11(3), 193-198. doi: 10.22061/jdma.2026.12640.1177
‎Ebrahim Vatandoost; ‎Ahmad Abbasi; Kamran Mirasheh. "Proof of a conjecture for the watching number". فناوری آموزش, 11, 3, 1405, 193-198. doi: 10.22061/jdma.2026.12640.1177
Vatandoost, ‎Ebrahim, Abbasi, ‎Ahmad, Mirasheh, Kamran. (1405). 'Proof of a conjecture for the watching number', فناوری آموزش, 11(3), pp. 193-198. doi: 10.22061/jdma.2026.12640.1177
Vatandoost, ‎Ebrahim, Abbasi, ‎Ahmad, Mirasheh, Kamran. Proof of a conjecture for the watching number. فناوری آموزش, 1405; 11(3): 193-198. doi: 10.22061/jdma.2026.12640.1177

Proof of a conjecture for the watching number

Journal of Discrete Mathematics and Its Applications
دوره 11، شماره 3، آذر 2026، صفحه 193-198    اصل مقاله (332.2 K)
نوع مقاله: Full Length Article
شناسه دیجیتال (DOI): 10.22061/jdma.2026.12640.1177
نویسندگان
‎Ebrahim Vatandoost1؛ ‎Ahmad Abbasi2؛ Kamran Mirasheh* 3
1‎Department of Basic Science, ‎Imam Khomeini International University‎‎,‎Qazvin‎, ‎Iran
2‎Department of Pure Mathematics, ‎Faculty of Mathematical Sciences‎, ‎University of Guilan‎‎, ‎Rasht‎, ‎Iran
3Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran
تاریخ دریافت: 30 مهر 1404،  تاریخ بازنگری: 20 بهمن 1404،  تاریخ پذیرش: 18 اسفند 1404 
چکیده
In this paper, we prove that for every connected graph G of order n, the watching number of its subdivision graph S(G) is exactly n. This result confirms a conjecture previously proposed by Vatandoost et al. [7].
کلیدواژه‌ها
watching system؛ subdivision؛ graph
مراجع
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[2] D. Auger, I. Charon, O. Hudry, A. Lobstein, Maximum size of a minimum watching system and the graphs achieving the bound, Discrete Applied Mathematics 164 (2014) 20–33. https://doi.org/10.1016/j.dam.2012.08.028

[3] F. Foucaud, S. Gravier, R. Naserasr, A. Parreau, P. Valicov, Identifying codes in line graphs, Journal of Graph Theory 73 (2013) 425–448. https://doi.org/10.1002/jgt.21686

[4] F. Foucaud, G. Perarnau, Bounds for identifying codes in terms of degree parameters, Electronic Journal of Combinatorics 19 (2012) P32. https://doi.org/10.37236/2036

[5] M. D. Hernando, M. Mora, I. M. Pelayo, Watching systems in complete bipartite graphs, VIII Jornadas de Matemática Discreta y Algortímica, Almería (2012) 53–60.

[6] O. Hudry, A. Lobstein, Unique (optimal) solutions: Complexity results for identifying and locating dominating codes, Theoretical Computer Science 767 (2019) 83–102. https://doi.org/10.1016/j.tcs.2018.09.034

[7] K. Mirasheh, A. Abbasi, E. Vatandoost, Domination number and watching number of subdivision construction of graphs, Facta Universitatis, Series: Mathematics and Informatics 40 (2025) 33–43. https://doi.org/10.22190/FUMI221126003M

[8] M. Roozbayani, H. Maimani, A. Tehranian, Watching systems of triangular graphs, Transactions on Combinatorics 3 (2014) 51–57. https://doi.org/10.22108/toc.2014.4127 
 [9] M. Roozbayani, H. R. Maimani, Identifying codes and watching systems in Kneser graphs, Discrete Mathematics, Algorithms and Applications 9 (2017) 1750007. https://doi.org/10.1142/S1793830917500070

[10] D. F. Rall, K. Wash, Identifying codes of the direct product of two cliques, European Journal of Combinatorics 36 (2014) 159–171. https://doi.org/10.1016/j.ejc.2013.07.002

[11] A. Shaminejad, E. Vatandoost, K. Mirasheh, The identifying code number and Mycielski’s construction of graphs, Transactions on Combinatorics 11 (2022) 309–316. https://doi.org/10.22108/toc.2021.126368.1794 

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