The Upper and Forcing Fault Tolerant Geodetic Number of a Graph
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
Keywords
Full Text:
PDFReferences
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.
DOI: http://dx.doi.org/10.23755/rm.v44i0.903
Refbacks
- There are currently no refbacks.
Copyright (c) 2022 T Jeba Raj, K Bensiger
This work is licensed under a Creative Commons Attribution 4.0 International License.
Ratio Mathematica - Journal of Mathematics, Statistics, and Applications. ISSN 1592-7415; e-ISSN 2282-8214.