Nothing Special   »   [go: up one dir, main page]
Skip to main content
SpringerLink
Log in

Domination integrity and efficient fuzzy graphs

  • Original Article
  • Published:
Neural Computing and Applications Aims and scope Submit manuscript

Abstract

In this paper, domination integrity of fuzzy graph and efficient fuzzy graph concepts is introduced with examples. An algorithm is developed to find whether an arc is strong or not. If it is strong, another algorithm will classify it as \(\alpha\) strong arc and \(\beta\) strong arc. The next algorithm is used to find whether the given fuzzy graph is a fuzzy tree or not. Domination and integrity are two different parameters used to define the stability of a graph in various situations. Using the strong arc concept a new parameter, domination integrity is defined and lower and upper bounds are found. This paper discusses the domination integrity for standard graphs such as path, cycle and complete graph. The domination integrity for Cartesian product of fuzzy graphs is also discussed. Finally, the new class of fuzzy graph, efficient fuzzy graph, is introduced. Efficient fuzzy graph is a special type of fuzzy graph that has the same dominating set, other than vertex set V, for both fuzzy graph and its underlying crisp graph.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Explore related subjects

Discover the latest articles, news and stories from top researchers in related subjects.

References

  1. Ore O (1962) Theory of graphs. American Mathematical Society, Providence

    MATH  Google Scholar 

  2. Barefoot CA, Entringer R, Swart H (1987) Vulnerability in graphs—a comparative survey. J Combin Math Combin Comput 1:12–22

    MathSciNet  MATH  Google Scholar 

  3. Barefoot CA, Entringer R, Swart H (1987) Integrity of trees and powers of cycles. Congr Numer 58:103–114

    MathSciNet  MATH  Google Scholar 

  4. Goddard W (1989) On the vulnerability of graphs, Ph.D. Thesis, University of Natal, Durban, SA

  5. Goddard W, Swart HC (1990) Integrity in graphs: bounds and basics. J Combin Math Combin 7:139–151

    MathSciNet  MATH  Google Scholar 

  6. Goddard W, Swart HC (1988) On the integrity of combinations of graphs. J Combin Math Combin Comput 4:3–18

    MathSciNet  MATH  Google Scholar 

  7. Bagga KS, Beineke LW, Goddard WD, Lipman MJ, Pipert RE (1992) A survey of integrity. Discret Appl Math 37(38):13–28

    MathSciNet  MATH  Google Scholar 

  8. Dundar P, Aytac A (2004) Integrity of total graphs via certain parameters. Math Notes 75(5):665–672

    MATH  Google Scholar 

  9. Mamut A, Vumar E (2007) A note on the integrity of middle graphs. Lect Notes Comput Sci 4381:130–134

    MathSciNet  MATH  Google Scholar 

  10. Mahde SS, Mathad V, Sahal AM (2015) Hub-integrity of graphs. Bull Int Math Virtual Inst 5:57–64

    MathSciNet  MATH  Google Scholar 

  11. Bagga KS, Beineke LW, Lipman MJ, Pippert RE (1994) Edge-integrity: a survey. Discret Math 124:3–12

    MathSciNet  MATH  Google Scholar 

  12. Li Y, Zhang S, Li X (2005) Rupture degree of graphs. Int J Comput Math 82(7):793–803

    MathSciNet  MATH  Google Scholar 

  13. Jung HA (1978) On a class of posets and the corresponding comparability graphs. J Combin Theory Ser B 24(2):125–133

    MathSciNet  MATH  Google Scholar 

  14. Cozzens M, Moazzami D, Stueckle S (1992) The tenacity of a graph. Graph theory, combinatorics, and algorithms, vol 1, 2. Wiley, New York, pp 1111–1122

    MATH  Google Scholar 

  15. Bauer D, Broersma H, Schmeichel E (2006) Toughness in graphs: a survey. Graphs Combin 22:1–35. https://doi.org/10.1007/s00373-006-0649-0

    Article  MathSciNet  MATH  Google Scholar 

  16. Sundareswaran R, Swaminathan V (2009) Domination integrity in graphs. Proc Int Conf Math Exp Phys Prague 3–8:46–57

    MATH  Google Scholar 

  17. Sundareswaran R, Swaminathan V (2010) Domination integrity of middle graphs. In: Chelvam T, Somasundaram S, Kala R (eds) Algebra, graph theory and their applications. Narosa Publishing House, New Delhi, pp 88–92

    Google Scholar 

  18. Sundareswaran R, Swaminathan V (2010) Domination integrity in trees. Bull Int Math Virtual Inst 2:153–161

    MathSciNet  MATH  Google Scholar 

  19. Sundareswaran R, Swaminathan V (2011) Domination integrity of powers of cycles. Int J Math Res 3(3):257–265

    Google Scholar 

  20. Sundareswaran R, Swaminathan V (2016) Integrity and domination integrity of gear graphs. J Appl Eng Math 6(1):54–64

    MathSciNet  MATH  Google Scholar 

  21. Sampathkumar E (1989) The global domination number of a graph. J Math Phys Sci 23:377–385

    MathSciNet  MATH  Google Scholar 

  22. Mahde SS, Mathad V (2017) Global domination integrity of graphs. Math Sci Lett 6:263–269

    MATH  Google Scholar 

  23. Zadeh LA (1965) Fuzzy sets. Inf Control 8(3):338–353

    MATH  Google Scholar 

  24. Rosenfeld A (1975) Fuzzy graphs. In: Zadeh LA, Fu KS, Shimura M (eds) Fuzzy sets and their applications. Academic press, New york, pp 77–95

    Google Scholar 

  25. Bhutani KR, Rosenfeld A (2003) Strong arcs in fuzzy graphs. Inf Sci 152:319–322

    MathSciNet  MATH  Google Scholar 

  26. Mathew S, Sunitha MS (2009) Types of arcs in a fuzzy graphs. Inf Sci 179:1760–1768

    MathSciNet  MATH  Google Scholar 

  27. Somasundaram A, Somasundaram S (1998) Domination in fuzzy graphs-I. Pattern Recognit Lett 19:787–791

    MATH  Google Scholar 

  28. Somasundaram A (2005) Domination in fuzzy graphs—II. J Fuzzy Math 13(2):281–288

    MathSciNet  MATH  Google Scholar 

  29. Nagoorgani A, Chandrasekaran VT (2006) Domination in fuzzy graph. Adv Fuzzy Sets Syst I(1):17–26

    MathSciNet  MATH  Google Scholar 

  30. Nagoorgani A, Vijayalakshmi P (2011) Insensitive arc in domination of fuzzy graph. Int J Contemp Math Sci 6(26):1303–1309

    MathSciNet  MATH  Google Scholar 

  31. Manjusha OT, Sunitha MS (2014) Notes on domination in fuzzy graphs. J Intell Fuzzy Syst 27(6):3205–3212. https://doi.org/10.3233/IFS-141277

    Article  MathSciNet  MATH  Google Scholar 

  32. Kalathodi S, Sunitha MS (2012) Distance in fuzzy graphs. LAP LAMBERT Academic Publishing, New York

    Google Scholar 

  33. Manjusha OT, Sunitha MS (2015) Strong domination in fuzzy graphs. Fuzzy Int Eng 7:369–377

    MathSciNet  MATH  Google Scholar 

  34. Saravanan M, Sujatha R, Sundareswaran R (2016) Integrity of fuzzy graphs. Bull Int Math Virtual Inst 6:89–96

    MathSciNet  MATH  Google Scholar 

  35. Saravanan M, Sujatha R, Sundareswaran R (2015) A study of regular fuzzy graphs and integrity of fuzzy graphs. Int J Appl Eng Res 10(82):160–164

    MATH  Google Scholar 

  36. Saravanan M, Sujatha R, Sundareswaran R (2018) Concept of integrity and its result in fuzzy graphs. J Intell Fuzzy Syst 34(4):2429–2439

    Google Scholar 

  37. Mariappan S, Ramalingam S, Raman S, Muthuselvan B (2018) Application of domination integrity of graphs in PMU placement in electric power networks. Turk J Electric Eng Comput Sci 26(4):2066–2076

    Google Scholar 

  38. Samanta S, Pal M (2015) Fuzzy planar graphs. IEEE Trans Fuzzy Syst 23(6):1936–1942. https://doi.org/10.1109/TFUZZ.2014.2387875

    Article  Google Scholar 

  39. Samanta S, Pal M (2013) Fuzzy k-competition graphs and p-competition fuzzy graphs. Fuzzy Inf Eng 5(5):191–204. https://doi.org/10.1007/s12543-013-0140-6S

    Article  MathSciNet  MATH  Google Scholar 

  40. Samanta S, Pal M (2012) Bipolar fuzzy hypergraphs. Int J Fuzzy Logic Syst 2(1):17–28

    Google Scholar 

  41. Samanta S, Akram M, Pal M (2015) m-step fuzzy competition graphs. J Appl Math Comput 47:461–472. https://doi.org/10.1007/s12190-014-0785-2

    Article  MathSciNet  MATH  Google Scholar 

  42. Rashmanlou H, Pal M (2013) Isometry on interval-valued fuzzy graphs. Int J Fuzzy Math Arch 3:28–35

    Google Scholar 

  43. Rashmanlou H, Pal M (2013) Antipodal interval-valued fuzzy graphs. Int J Appl Fuzzy Sets Artif Intell 3:107–130

    Google Scholar 

  44. Pal M, Rashmanlou H (2013) Irregular interval—valued fuzzy graphs. Ann Pure Appl Math 3(1):56–66

    Google Scholar 

  45. Mathew S, Sunitha MS (2010) Node connectivity and arc connectivity of a fuzzy graph. Inf Sci 180(4):519–531

    MathSciNet  MATH  Google Scholar 

  46. Sensarma D, Sen Sarma S (2019) Role of graphic integer sequence in the determination of graph integrity. Mathematics 7:261

    Google Scholar 

  47. Kalathian S, Ramalingam S, Srinivasan N, Raman S, Broumi S (2019) Embedding of fuzzy graphs on topological surfaces. Neural computing and applications. Springer, New York, pp 1–11

    Google Scholar 

  48. Halim Z, Khattak JH (2018) Density-based clustering of big probabilistic graphs. Evolving systems. Springer, New York, pp 1–18

    Google Scholar 

  49. Rashid A, Kamran M, Halim Z (2019) A top down approach to enumerate \(\alpha\)-maximal cliques in uncertain graphs. J Intell Fuzzy Syst 1–13 (Preprint)

  50. Halim Z, Waqas M, Baig AR, Rashid A (2017) Efficient clustering of large uncertain graphs using neighborhood information. Int J Approx Reason 90:274–291

    MathSciNet  MATH  Google Scholar 

  51. Jamil S, Khan A, Halim Z, Baig AR (2011) Weighted muse for frequent sub-graph pattern finding in uncertain DBLP data. In: 2011 international conference on internet technology and applications, IEEE, pp 1–6

  52. Sunitha MS, Vijayakumar A (1999) A characterization of fuzzy trees. Inf Sci 113:293–300

    MathSciNet  MATH  Google Scholar 

  53. Tom M, Sunitha MS (2015) Strong sum distance in fuzzy graphs. Springerplus 4:214. https://doi.org/10.1186/s40064-015-0935-5

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank the Management and the Principal, SSN College of Engineering, OMR, Chennai, and Mannar Thirumalai Naicker College, Pasumalai, Madurai.

Author information

Authors and Affiliations

  1. Department of Mathematics, Mannar Thirumalai Naicker College, Pasumalai, Madurai, Tamil Nadu, India

    Saravanan Mariappan

  2. Department of Mathematics, SSN College of Engineering, Old Mahabalipuram Road, Chennai, Tamil Nadu, India

    Sujatha Ramalingam & Sundareswaran Raman

  3. Department of Mathematics, Celal Bayar University, 45140, Manisa, Turkey

    Goksen Bacak-Turan

Authors
  1. Saravanan Mariappan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Sujatha Ramalingam
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Sundareswaran Raman
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Goksen Bacak-Turan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sujatha Ramalingam.

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

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Mariappan, S., Ramalingam, S., Raman, S. et al. Domination integrity and efficient fuzzy graphs. Neural Comput & Applic 32, 10263–10273 (2020). https://doi.org/10.1007/s00521-019-04563-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00521-019-04563-5

Keywords

Search

Navigation