Abstract

The (G,D)-bondage number of a graph G denoted by b𝛾𝐺 (𝐺) is the least positive integer k such that there exists 𝐹 ⊆ 𝐸(𝐺) with 𝐹 = 𝑘 and 𝛾𝐺 (𝐺 − 𝐹) > 𝛾𝐺 (𝐺). If no such k exists, it is defined to be ∞. The (G,D)-nonbondage number of a graph G denoted by bn𝛾𝐺 (𝐺) is defined as the maximum cardinality among all sets of edges 𝑋 ⊆ 𝐸(𝐺) such that 𝛾𝐺 𝐺 − 𝑋 = 𝛾𝐺 𝐺 . If bn𝛾𝐺 (𝐺)does not exist, we define bn𝛾𝐺 𝐺 = 0.In this paper we initiate a study of these two parameters.

Keywords

References

[1] Buckley F, Harary F and Quintas V L, Extremal results on the geodetic number of a graph, Scientia, volume A2 (1988), 17-26.
[2] Chartrand G, Harary F and Zhang P, Geodetic sets in graphs, DiscussionesMathematicae Graph theory, 20 (2000), 129-138e.
[3] Chartrand G, Harary F and Zhang P, On the Geodetic number of a graph, Networks, Volume 39(1) (2002), 1-6.
[4] Chartrand G, Zhang P and Harary F, Extremal problems in Geodetic graph Theory, CongressusNumerantium 131 (1998), 55-66.
[5] Fink J F, Jacobson M S, Kinch I F and Roberts J, The bondage number of a graph, Discrete Math., 86(1-3), (1990), 47-57.
[6] Haynes T W, Hedetniemi S T and Slater P J, Fundamentals of Domination in Graphs, Marcel Decker Inc., 1998.
[7] Kulli V R and Janakiram B, The nonbondage number of a graph, Graph Theory Notes of New York, New York Academy of Sciences, 30 (1996) 14-16.
[8] Palani K and Nagarajan A, (G,D)-Number of a graph, International Journal of Mathematics Research, Volume 3, Number 3 (2011), 285-299.
[9] Palani K and Kalavathi S, (G,D)-Number of some special graphs, International Journal of Engineering and Mathematical Sciences, January-June 2014, Volume 5, Issue-1, pp.7-15, ISSN(Print) - 2319 – 4537, (Online) – 2319 – 4545.
[10] Palani K , Nagarajan A and Mahadevan G, Results connecting domination, geodetic and (G,D)-number of graph, International Journal of Combinatorial graph theory and applications, Volume 3, No.1, January – June (2010)(pp.51 -59).