The Upper and Forcing Fault Tolerant Geodetic Number of a Graph

Authors

  • T Jeba Raj Assistant Professor, Department of Mathematics, Malankara Catholic College, Mariagiri, Kaliyakkavilai - 629 153, India,
  • K Bensiger Register Number. 20123082091004, Research Scholar, Department of Mathematics, Malankara Catholic College, Mariagiri, Kaliyakkavilai - 629 153, India

DOI:

https://doi.org/10.23755/rm.v44i0.903

Keywords:

tolerant geodetic, connected graphs, minimum cardinality

Abstract

A fault tolerant geodetic  is said to be minimal fault tolerant geodetic set of  if no proper subset of  is a fault tolerant geodetic set of  is called the upper fault tolerant geodetic number of  is denoted by . Some general properties satisfied by this concept are studied. For connected graphs of order  with to be  is given. It is shown that for every pair of with , there exists a connected graph such that  and , where  is the fault tolerant geodetic number of  and  is the upper fault tolerant geodetic number of a graph. Let S be a -set of . A subset  is called a forcing subset for  if  is the unique -set containing T. A forcing subset for  of minimum cardinality is a minimum forcing subset of . The forcing fault tolerant geodetic number of S, denoted by, is the cardinality of a minimum forcing subset of . The forcing fault tolerant geodetic number of , denoted by  is , where the minimum is taken over all -sets in . The forcing fault tolerant geodetic number of some standard graphs are determined. Some of its general properties are studied. It is shown that for every pair of positive integers  and  with  and  there exists a connected graph  such that  and

References

H.A. Ahangar, S. Kosari, S.M. Sheikholeslami and L. Volkmann, Graphs with large geodetic number, Filomat, 29:6 (2015), 1361 – 1368.

H. AbdollahzadehAhangar, V. Samodivkin, S. M. Sheikholeslami and Abdollah Khodkar, The restrained geodetic number of a graph, Bulletin of the Malaysian Mathematical Sciences Society, 38(3), (2015), 1143-1155.

H. Abdollahzadeh Ahangar, Fujie-Okamoto, F. and Samodivkin, V., On the forcing connected geodetic number and the connected geodetic number of a graph, Ars Combinatoria, 126, (2016), 323-335.

H. AbdollahzadehAhangar and Maryam Najimi, Total Restrained Geodetic Number of Graphs, Iranian Journal of Science and Technology, Transactions A: Science, 41, (2017), 473–480.

F. Buckley and F. Harary, Distance in Graphs, Addition- Wesley, Redwood City, CA, (1990). [6] G. Chartrand, P. Zhang, The forcing geodetic number of a graph, Discuss. Math. GraphTheory, 19 (1999), 45–58.

G. Chartrand, F. Harary and P. Zhang, “On the geodetic number of a graph”, Networks, 39(1), (2002), 1 - 6.

H. Escaudro, R. Gera, A. Hansberg, N. Jafari Rad and L. Volkmann,” Geodetic domination in graphs”, Journal of Combinatorial Mathematics and Combinatorial Computing, 77, (2011), 88- 101.

T.W. Hayes, P.J. Slater and S.T. Hedetniemi, Fundamentals of domination in graphs, Boca Raton, CA: CRC Press, (1998).

A. Hansberg and L. Volkmann, On the geodetic and geodetic domination numbers of a graph, Discrete Mathematics, 310, (2010), 2140-2146.

Mitre C. Dourado, Fabio Protti, Dieter Rautenbach and Jayme L. Szwarcfiter, Some remarks on the geodetic number of a graph, Discrete Mathematics,310,(2010), 832-837.

H.M. Nuenay and F.P. Jamil,” On minimal geodetic domination in graphs”,

Discussiones Mathematicae graph Theory, 35,(3), (2015), 403-418.

Downloads

Published

2022-12-11

Issue

Vol. 44 (2022): Special Issue: Recent Trends in Mathematics and its Applications

Section

Volume 45-2023

License

This journal provides immediate open access to its content on the principle that making research freely available to the public supports a greater global exchange of knowledge. The journal is committed to real and immediate open access for academic work. All of the journal articles are free to access immediately from the date of publication. There are no charge for readers to download articles and reviews for their own scholarly use. Benefits of open access for authors, include:

  • Free access for all users worldwide
  • Increased visibility and readership
  • Rapid publication
  • No spatial constraints
  • Authors can share their research papers on their personal web-pages (i.e. ResearchGate, etc.)

Ratio Mathematica publishes open access articles under the terms of the Creative Commons Attribution (CC BY) License. The Creative Commons Attribution License (CC-BY) allows users to copy, distribute and transmit an article, adapt the article and make commercial use of the article. The CC BY license permits commercial and non-commercial re-use of an open access article, as long as the author is properly attributed. Copyright on any research article published by Ratio Mathematica is retained by the author(s). Authors grant Ratio Mathematica a license to publish the article and identify itself as the original publisher. Authors also grant any third party the right to use the article freely as long as its original authors, citation details and publisher are identified. 
For further information visit: https://creativecommons.org/licenses/by/3.0/. The following logo will appear in any paper that will be published from the volume number 31/2016 onwards.