Abstract
In graph theory, a vertex covering set \(V_\mathrm{C}\) is a set of vertices such that each edge of the graph is incident to at least one of the vertices of the set \(V_\mathrm{C}\). The problems related to vertex covering are called vertex covering problems. Many real-life problems contain a lot of uncertainties. To handle such uncertainties, concept of fuzzy set/graph is used. Here, we consider the covering problems of fuzzy graph to model some real-life problems. In this paper, a vertex covering problem is modeled as a series of linear and nonlinear programming problems with the help of basic graph-theoretic concept. In this model, the following objectives are considered: (1) the total number of facilities, the coverage area and total efficiency of all facilities are maximized, whereas (2) the total cost for the covering problem is minimized. Some new sets are defined and determined to make best decision on the basis of the features of facilities of the fuzzy system. An illustration is given to describe the whole model. Application of the said vertex covering problem to make a suitable decision for the placement of CCTVs in a city with the help of the developed formulations is given in a systematic way. To find the solutions, some algorithms are designed and the mathematical software ‘LINGO’ is used to keep the fuzziness of the parameters involved in the problems.
Similar content being viewed by others
References
Blue MB, Bush B, Puckett J (2002) Unified approach to fuzzy graph problems. Fuzzy Sets Syst 125:355–368
Chang SSL, Zadeh L (1972) On fuzzy mappings and control. IEEE Trans Syst Man Cybern 2:30–4
Chaudhry SS (1993) New heuristics for the conditional covering problem. Opsearch 30:42–47
Chaudhry SS, Moon ID, McCormick ST (1987) Conditional covering: greedy heuristics and computational results. Comput Oper Res 14:11–18
Chen SJ, Chen SM (2001) A new method to measure the similarity between fuzzy numbers. In: IEEE international conference on fuzzy systems, pp 1123–1126
Dinur I, Safra S (2005) On the hardness of approximating minimum vertex cover. Ann Math 162(1):1–32
Dubois D, Prade H (1978) Operations on fuzzy numbers. Int J Syst Sci 9:613–626
Ghorai G, Pal M (2016) A study on m-polar fuzzy planar graphs. Int J Comput Sci Math 7(3):283–292
Ghorai G, Pal M (2016) Faces and dual of m-polar fuzzy planar graphs. J Intell Fuzzy Syst 31(3):2043–2049
Hastad J (2001) Some optimal in approximality results. J ACM 48(4):798–859
Kauffmann A (1980) Introduction to fuzzy arithmetic: theory and applications. Van Nostrand Reinhold, New York
Khot S, Regev O (2008) Vertex cover might be hard to approximate to within \(2-\in \). J Comput Syst Sci 74:335–349
Koczy LT (1992) Fuzzy graphs in the evaluation and optimization of networks. Fuzzy Sets Syst 46:307–319
Lotfi V, Moon ID (1997) Hybrid heuristics for conditional covering problem. Int J Model Simul 17:185–190
Lunday BJ, Smith JC, Gold-berg JB (2005) Algorithms for solving the conditional covering problem on paths. Naval Res Log 52:293–301
Moon ID, Chaudhry SS (1984) An analysis of network location problems with distance constraints. Manag Sci 30:290–307
Mordeson JN, Peng CS (1994) Operation on fuzzy graphs. Inf Sci 79:159–170
Nayeem SMA, Pal M (2005) Shortest path problem on a network with imprecise edge weight. Fuzzy Optim Decis Mak 4(4):293–312
Ni Y (2005) Models and algorithm for stochastic minimum weight edge covering problem. In: Proceedings of the fourth international conference on information and management sciences, Yunnan, China, pp 445–451
Ni Y (2008) Fuzzy minimum weight edge covering problem. Appl Math Model 32:1327–1337
Pathinathan T, Mike Dison E (2018) Similarity measures of pentagonal fuzzy numbers. Int J Pure Appl Math 119(9):165–175
Pathinathan T, Ponnivalavan K (2014) Pentagonal fuzzy Number. Int J Comput Algorithm 3:1003–1005
Pramanik T, Samanta S, Pal M (2016) Interval-valued fuzzy planar graphs. Int J Mach Learn Cybern 7(4):653–664
Pramanik T, Samanta S, Sarkar B, Pal M (2017) Fuzzy \(\phi \)-tolerance competition graphs. Soft Comput 21(13):3723–3734
Rashmanlou H, Pal M (2014) Isometry on interval-valued fuzzy graphs. arXiv preprint arXiv:1405.6003
Rashmanlou H, Samanta S, Pal M, Borzooei RA (2015) A study on bipolar fuzzy graphs. J Intell Fuzzy Syst 28(2):571–580
Rosenfield A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M (eds) Fuzzy sets and their application. Academic press, New York, pp 77–95
Sahoo S, Pal M (2016) Intuitionistic fuzzy competition graphs. J Appl Math Comput 52(1–2):37–57
Samanta S, Akram M, Pal M (2015) m-step fuzzy competition graphs. J Appl Math Comput 47(1–2):461–472
Samanta S, Pal M, Pal A (2014) New concepts of fuzzy planar graph. Int J Intell Adv Res Artif 3(1):52–59
Samanta S, Pramanik T, Pal M (2016) Fuzzy colouring of fuzzy graphs. Afr Math 27(1–2):37–50
Samanta S, Pal M (2011) Fuzzy tolerance graphs. Int J Latest Trends Math 1(2):57–67
Samanta S, Pal M (2013) Fuzzy information and engineering. In: Fuzzy k-competition graphs and p-competition fuzzy graphs, pp 1–14
Samanta S, Pal M (2015) Fuzzy planar graphs. IEEE Trans Fuzzy Syst 23(6):1936–1942
Saha A, Pal M, Pal TK (2007) Selection of programme slots of television channels for giving advertisement: A graph theoretic approach. Inf Sci 177(12):2480–2492
Yager RR (1996) Knowledge-based defuzzification. Fuzzy Sets Syst 80:177–185
Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353
Acknowledgements
Financial support of the first author offered by DHESTBT, Government of West Bengal (Memo No. \( 353(Sanc.)/ST/P/S \& T/16G-15/2018\)) is thankfully acknowledged.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Bhattacharya, A., Pal, M. Vertex covering problems of fuzzy graphs and their application in CCTV installation. Neural Comput & Applic 33, 5483–5506 (2021). https://doi.org/10.1007/s00521-020-05324-5
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00521-020-05324-5