Some Results on Strong Edge Geodetic Problem in Graphs

Authors

  • D. Antony Xavier Department of Mathematics, Loyola College (University of Madras), Chennai
  • Deepa Mathew Department of Mathematics, Loyola College (University of Madras), Chennai
  • Santiagu Theresal Department of Mathematics, Loyola College (University of Madras), Chennai
  • Eddith Sarah Varghese Department of Mathematics, Loyola College (The University of Madras), Chennai

DOI:

https://doi.org/10.26713/cma.v11i3.1385

Keywords:

Strong edge geodetic number, Strong geodetic number, Edge geodetic number, Geodetic set

Abstract

For a graph \(G(V(G),E(G))\), the problem to find a \(S\subseteq V(G)\) where every edge of the graph \(G\) is covered by a unique fixed geodesic between the pair of vertices in \(S\) is called the strong edge geodetic problem and the cardinality of the smallest such \(S\) is the strong edge geodetic number of \(G\). In this paper the strong edge geodetic problem for product graphs are studied and also some results for general graphs are derived.

Downloads

Download data is not yet available.

References

G. Chartrand, E. M. Palmer and P. Zhang, The geodetic number of a graph: a survey, in Proceedings of the Thirty-third Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 2002), Congr. Numer. 156 (2002), 37 – 58.

G. Chartrand, F. Harary and P. Zhang, On the geodetic number of a graph, Networks 39(1) (2002), 1 – 6, DOI: 10.1002/net.10007.

H. Ge, Z. Wang and J. Zou, Strong geodetic number in some networks, Journal of Mathematics Research 11(2) (2019), 20 – 29, DOI: 10.5539/jmr.v11n2p20.

V. Gledel and V. Iršiˇc, Strong geodetic number of complete bipartite graphs, crown graphs and hypercubes, Bulletin of the Malaysian Mathematical Sciences Society 43 (2020), 2757 – 2767, DOI: 10.1007/s40840-019-00833-6.

N. Goyal and M. Mali, A study of geodetic graphs, Conference Proceedings (2015).

V. Iršiˇc, Strong geodetic number of complete bipartite graphs and of graphs with specified diameter, Graphs and Combinatorics 34(3) (2018), 443 – 456, DOI: 10.1007/s00373-018-1885-9.

V. Iršiˇc and M. Konvalinka, Strong geodetic problem on complete multipartite graphs, Ars Mathematica Contemporanea 17(2) (2019), 481 – 491, DOI: 10.26493/1855-3974.1725.2e5.

V. Iršiˇc and S. Klavžar, Strong geodetic problem on Cartesian products of graphs, RAIRO-Operations Research 52(1) (2018), 205 – 216, DOI: 10.1051/ro/2018003.

S. Klavžar and P. Manuel, Strong geodetic problem in grid-like architectures, Bulletin of the Malaysian Mathematical Sciences Society 41(3) (2018), 1671 – 1680, DOI: 10.1007/s40840-018-0609-x.

P. Manuel, S. Klavžar, A. Xavier, A. Arokiaraj and E. Thomas, Strong edge geodetic problem in networks, Open Mathematics 15(1) (2017), 1225 – 1235, DOI: 10.1515/math-2017-0101.

P. Manuel, S. Klavžar, A. Xavier, A. Arokiaraj and E. Thomas, Strong geodetic problem in networks, Discussiones Mathematicae Graph Theory (2018), DOI: 10.7151/dmgt.2139.

I. M. Pelayo, Geodesic Convexity in Graphs, Springer, New York (2013), DOI: 10.1007/978-1-4614-8699-2.

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.

A. Santhakumaran and S. Ullas Chandran, The edge geodetic number and Cartesian product of graphs, Discussiones Mathematicae Graph Theory 30(1) (2010), 55 – 73, DOI: 10.7151/dmgt.1476.

A. P. Santhakumaran, T. Jebaraj and S. V. Ullas Chandran, Linear edge geodetic graphs, Journal of Applied Mathematics and Informatics 30(56) (2012), 871 – 882.

A. P. Santhakumaran, T. Jebaraj and S. V. Ullas Chandran, The linear geodetic number of a graph, Discrete Mathematics, Algorithms and Applications 3(3) (2011), 357 – 368, DOI: 10.1142/s1793830911001279.

Z.Wang, Y. Mao, H. Ge and C. Magnant, Strong geodetic number of graphs and connectivity, Bulletin of the Malaysian Mathematical Sciences Society 43 (2020), 2443 – 2453, DOI: 10.1007/s40840-019-00809-6.

A. Xavier, S. Theresal and D. Mathew, Strong shortest path union cover for certain graphs, Adalya Journal 9(2) (2020), 1086 – 1100, URL: https://drive.google.com/open?id=1t3t8o5PqZV2cCpv5W2S5gvk9HNN12eXm.

Downloads

Published

30-09-2020
CITATION

How to Cite

Xavier, D. A., Mathew, D., Theresal, S., & Varghese, E. S. (2020). Some Results on Strong Edge Geodetic Problem in Graphs. Communications in Mathematics and Applications, 11(3), 403–413. https://doi.org/10.26713/cma.v11i3.1385

Issue

Vol. 11 No. 3 (2020)

Section

Research Article

License

Creative Commons Licence 4.0 by  

Authors who publish with this journal agree to the following terms:

  • Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a CCAL that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
  • Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
  • Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work.