Characterizing Graphs via Edge Geodetic Domination Number

Authors

  • Arvind Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, Haryana, India
  • Seema Mehra Department of Mathematics, Maharshi Dayanand University, Rohtak 124001, Haryana, India

DOI:

https://doi.org/10.26713/cma.v17i1.3443

Keywords:

Edge geodetic dominating set, Edge geodetic dominance number, Antiprism graph alternate pentagonal snake graph, Bistar graph, Ladder graph

Abstract

Dominance in graphs is the crucial aspect of graph theory that has been thoroughly examined. Consider a graph \(G_1(V_1,E_1)\) with \(S\subseteq V_1\) in such a way that at least one vertex in the set is adjacent to the vertices that do not belong to the set then \(S\) is said to be the dominating set. In other words, it can be said that the set of vertices belonging to \(S\) and \(S'\) has at least a single neighbor in each other. Any set \(A\subseteq G_1\) having all edges of \(G\) comprised in a geodesic uniting a pair of vertices in \(A\) is claimed to be an EG-set of \(G_1\). The EG-number, indicated by the symbol \(g_e(G)\), is the lowest order of its EG-set. A \(g_e\)-set of \(G\) or EG-basis of \(G\), is any EG-set of order \(g_e(G)\). If a collection of vertices \(D\) in \(G\) is together an EG-set and a dominant set then \(D\) is considered an EG-dominating set. The EG-dominance number of the EG-dominating set is its minimum cardinal number represented as \(\gamma_{ge}(G)\). With this work, we explore the EG-dominance number of varied graphs namely antiprism graph \(A_n\), alternate pentagonal snake \(A(\PS_n)\), Bistar graph, ladder graph, jewel graph and Helm graph.

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References

[1] A. M. Asdain, J. I. C. Salim and R. G. Artes Jr., Geodetic bounds in graphs, International Journal of Mathematics and Computer Science 18(4) (2023), 767 – 771, URL: https://future-in-tech.net/18.4/R-RosalioArtesJr.pdf.

[2] C. Berge, The Theory of Graphs, Dover Publication, 272 pages (1958).

[3] S. Cheng, D. Wang and X. Liu, Hamiltonicity of Myceilski graphs, American Journal of Applied Mathematics 6(1) (2018), 20 – 22, DOI: 10.11648/j.ajam.20180601.14.

[4] H. Escuadro, R. Gera, A. Hansberg, N. R. Jafari and L. Volkmann, Geodetic domination in graphs, Journal of Combinatorial Mathematics and Combinatorial Computing 77 (2011), 89 – 101, URL: https://combinatorialpress.com/jcmcc-articles/volume-077/.

[5] A. Hansberg and L. Volkmann, On the geodetic and geodetic domination number of a graph, Discrete Mathematics 310(15-16) (2010), 2140 – 2146, DOI: 10.1016/j.disc.2010.04.013.

[6] T. W. Haynes, S. T. Hedetniemi and P. J. Slater, Fundamentals of Domination in Graphs, 1st edition, CRC Press, Boca Raton, 464 pages (1998), DOI: 10.1201/9781482246582.

[7] S. Leel, S. Srivastav, S. Gupta and G. Ganesan, Domination number in the context of some new graphs, Engineering Proceedings 62(1) (2024), 14, DOI: 10.3390/engproc2024062014.

[8] B. Mohamed and M. Badawy, Some new results on domination and independent dominating set of some graphs, Applied and Computational Mathematics 13(3) (2024), 53 – 57, DOI: 10.11648/j.acm.20241303.11.

[9] C. J. M. Quije, R. E. Mariano and E. C. Ahmad, Edge geodetic dominating sets of some graphs, European Journal of Pure and Applied Mathematics 18(1) (2025), Article number 5555, DOI: 10.29020/nybg.ejpam.v18i1.5555.

[10] A. P. Santhakumaran and J. John, Edge geodetic number of a graph, Journal of Discrete Mathematical Sciences and Cryptography 10(3) (2007), 415 – 432, DOI: 10.1080/09720529.2007.10698129.

[11] D. Stalin and J. John, Edge geodetic domination in graphs, International Journal of Pure and Applied Mathematics 116(22) (2017), 31 – 40, URL: https://acadpubl.eu/jsi/2017-116-13-22/articles/22/4.pdf.

[12] P. A. P. Sudhahar, A. Ajitha and A. Subramanian, The total edge geodetic domination number of a graph, South East Asian Journal of Mathematics and Mathematical Sciences 13(1) (2017), 19 – 26, URL: https://rsmams.org/journals/seajmams/article/183.

[13] D. B. West, Introduction to Graph Theory, Prentice Hall, 512 pages (1996).

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Published

30-03-2026

Issue

Vol. 17 No. 1 (2026)

Section

Research Article

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How to Cite

Arvind, & Mehra, S. (2026). Characterizing Graphs via Edge Geodetic Domination Number. Communications in Mathematics and Applications, 17(1), 163-174. https://doi.org/10.26713/cma.v17i1.3443