Geodetic Domination Integrity of Thorny Graphs

Abstract

The concept of geodetic domination integrity is a crucial parameter when examining the potential damage to a network. It has been observed that the removal of a geodetic set from the network can increase its vulnerability. This study explores the geodetic domination integrity parameter and presents general results on the geodetic domination integrity values of thorn ring graphs, $n$-sunlet graphs, thorn path graphs, thorn rod graphs, thorn star graphs, helm graphs, $E_p^t$ tree graphs, dendrimer graphs, spider graphs, and bispider graphs, which are the frequently used graph classes in the literature.

Keywords

• Geodetic domination integrity
• geodetic dominating set
• geodetic set
• dominating set

References

1. T. W. Haynes, S. Hedetniemi, P. Slater, Fundamentals of domination in graphs, CRC Press, Boca Raton, 1998.
2. C. A. Barefoot, R. Entringer, H. Swart, Vulnerability in graphs - A comparative survey, Journal of Combinatorial Mathematics and Combinatorial Computing 1 (1987) 13-22.
3. R. Sundareswaran, V. Swaminathan, Domination integrity of middle graphs, in: T. Tamizh Chelvam, S. Somasundaram, R. Kala (Eds.), Algebra, Graph Theory and Their Applications, Narosa Publishing House, New Delhi, 2010, pp. 88-92.
4. R. Sundareswaran, V. Swaminathan, Domination integrity in trees, Bulletin of International Mathematical Virtual Institute 2 (2012) 153-161.
5. G. Balaraman, S. S. Kumar, R. Sundareswaran, Geodetic domination integrity in graphs, TWMS Journal of Applied and Engineering Mathematics 11 (Special Issue) (2021) 258-267.
6. F. Buckley, F. Harary, L. V. Quintas, Extremal results on the geodetic number of a graph, Scientia A (2) (1988) 17-26.
7. F. Harary, Graph theory, Addison Wesley Publishing Company, New York, 1969.
8. J. A. Bondy, U. S. R. Murty, Graph theory with applications, Macmillan, London, 1976.

9. G. Chartrand, L. Lesniak, P. Zhang, Graphs digraphs, 4th Edition, Chapman and Hall/CRC, New York, USA, 2015.
10. T. H. Cormen, C. E. Leiserson, R. L. Rivest, C. Stein, Introduction to algorithms, The MIT Press, Cambridge, 2022.
11. H. Escuadro, R. Gera, A. Hansberg, N. Jafari Rad, L. Volkmann, Geodetic domination in graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 77 (2011) 89-101.
12. M. Azari, On the Gutman index of Thorn graphs, Kragujevac Journal of Science 40 (2018) 33-48.
13. A. Shobana, B. Logapriya, Domination number of n-sunlet graph, International Journal of Pure and Applied Mathematics 118 (20) (2018) 1149-1152.
14. I. Gutman, Distance of thorny graph, Publications de l'Institut Mathématique 63 (77) (1998) 31-36.
15. P. Shiladhar, A. M. Naji, N. D. Soner, Leap Zagreb indices of some wheel related graphs, Journal of Computer and Mathematical Sciences 9 (3) (2018) 221-231.
16. A. K. Nagar, S. Sriram, On eccentric connectivity index of eccentric graph of regular dendrimer, Mathematics in Computer Science 10 (2) (2016) 229-237.
17. B. Sahin, A. Sahin, On dominaton type invariants of regular dendrimer, Journal of Discrete Mathematics and Its Applications 7 (3) (2022) 147-152.
18. N. B. Ibrahim, A. A. Jund, Edge connected domination polynomial of a graph, Palestine Journal of Mathematics 7 (2) (2018) 458-467.