2-Domination in vague graphs

Document Type : Research Paper


1 Department of Knowledge and Cognitive Intelligence, Imam Hossein University, Tehran, Iran.

2 Department of Mathematics, Shahid Beheshti University, Tehran, Iran.

3 Department of Mathematics, Faculty of Science, Payam Noor University, Tehran, Iran.



A vague graph is a generalized structure of a fuzzy graph that gives more precision, flexibility and compatibility to a system when compared with systems that are designed using fuzzy graphs. In this paper, the notions of (perfect-total) 2-dominating set and (perfect-total) 2-domination numbers on vague graphs are introduced and some properties are investigated. Especially, it is proven that in any strong vague graph on a Petersen graph, any minimal 2-dominating set is a minimal perfect 2-dominating set and minimal dominating set. Then, the concepts of (total) 2-cobondage set and (total) 2-cobondage number in vague graphs are expressed and related results obtained. Finally, an application related to Fire Stations and Emergency Medical centers is provided.


[1] M. Akram, Certain types of vague cycles and vague trees, J. Intell. Fuzzy Syst., 28 (2015) 621-631.
[2] M. Akram, W. J. Chen and K. P. Shum, Some properties of vague graphs, Southeast Asian Bull. Math., 37 (2013) 307-324.
[3] M. Akram, W. A. Dudek and M. M. Yousaf, Regularity in vague intersection graphs and vague line graphs, In Abstract and Applied Analysis (Vol. 2014), Hindawi.
[4] M. Akram, A. Farooq, A. B. Saeid, and K. P. Shum, Certain types of vague cycles and vague trees, J. Intell. Fuzzy Syst., 28 No. 2 (2015) 621-631.
[5] S. Banitalebi and R. A. Borzooei Domination of vague graphs by using of strong arcs, submitted.
[6] R. A. Borzooei and H. Rashmanlou, Domination in vague graph and its application, J. Intell. Fuzzy Syst., 29 (2015) 1933-1940.
[7] R. A. Borzooei and H. Rashmanlou, Degree of vertices in vague graphs, J. Appl. Math. Informatics, 33 (2015) 545-557.
[8] R. A. Borzooei and H. Rashmanlou, S. Samanta and M. Pal, Regularity of vague graphs, J. Intell. Fuzzy Syst., 30 No. 6 (2016) 3681-3689.
[9] E. J. Cockayne and S. Hedetniem, Towards a theory of domination in graphs, Networks, 7 (1977) 247-261.
[10] W. L. Gau and D. J. Buehrer, Vague sets, IEEE Trans. Syst. Man Cybern. Syst., 23 No. 2 (1993) 610-614.
[11] B. S. Hoseini, M. Akram, M. S. Hoseini, H. Rashmanlou and R. A. Borzooei, Maximal product of graphs under vague environment, Math. Comput. Appl., 25 No. 1 (2020) 10.
[12] V. R. Kulli and B. Janakiram, The cobondage number of a graph, Discuss. Math. Graph Theory, 16 No. 2 (1996) 111-117.
[13] A. Nagoor Gani and V. T. Chandrasekaran, Domination in fuzzy graph, Adv. Fuzzy Syst., 1 No. 1 (2006) 17-26.
[14] A. Nagoor Gani and K. Prasanna Devi, 2-domination in fuzzy graphs, Int. J. Fuzzy Syst., 1 (2015) 119-124.
[15] A. Nagoor Gani and K. Prasanna Devi, Reduction of domination parameter in fuzzy graphs, Glob. J. Pure Appl. Math., 13 No. 7 (2017) 3307-3315.
[16] R. Parvathi, G. Thamizhendhi, Domination in intuitionistic fuzzy graphs, Fourteenth International Conference on Intuitionistic Fuzzy Sets, Sofia, 16 No. 2 (2010) 39-49.
[17] N. Ramakrishna, Vague graphs, International Journal of Computational Cognition, 7 (2009) 51-58.
[18] H. Rashmanlou and R. A. Borzooei, Product vague graphs and its applications, J. Intell. Fuzzy Syst., 30 No. 1 (2016) 371-382.
[19] H. Rashmanlou, S. Samanta, M. Pal and R. A. Borzooei, Product of bipolar fuzzy graphs and their degree, Int. J. Gen. Syst., 45 No. 1 (2016) 1-14.
[20] A. Rosenfeld, Fuzzy graphs, Fuzzy Sets and their Applications, (L. A. Zadeh, K. S. Fu and M. Shimura, Eds.), Academic Press, New York, (1975) 77-95.
[21] A. Somasundaram and S. Somasundaram, Domination in fuzzy graphs-I, Pattern Recognit. Lett., 19 (1998) 787-791.
[22] L. A. Zadeh, Fuzzy sets, Inf. Control., 8 (1965) 338-353.