Bounds on Co-Secure Domination in Graphs

Aleena Joseph; V.Sangeetha

doi:https://doi.org/10.14445/22315373/IJMTT-V55P520

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 55 | Number 2 | Year 2018 | Article Id. IJMTT-V55P520 | DOI : https://doi.org/10.14445/22315373/IJMTT-V55P520

Bounds on Co-Secure Domination in Graphs

Aleena Joseph, V.Sangeetha

Citation :

Aleena Joseph, V.Sangeetha, "Bounds on Co-Secure Domination in Graphs," International Journal of Mathematics Trends and Technology (IJMTT), vol. 55, no. 2, pp. 158-164, 2018. Crossref, https://doi.org/10.14445/22315373/IJMTT-V55P520

Abstract

Let G = (V, E) be a graph and let S⊆V. The set S is a co-secure dominating set (CSDS) of a graph G if S is a dominating set, and for each u ∈S there exists a vertex v ∈ V \ S such that uv ∈E(G) and (S \ {u})∪ {v} is a dominating set. The minimum cardinality of a co-secure dominating set in G is the co-secure domination number 𝛾𝑐𝑠 (G). We determine the co-secure domination number of some families of standard graphs and obtain sharp bounds. A set S ⊆V is a secure dominating set of a graph G = (V, E),if for each u ∈V \ S there exists a vertex v ∈ S such that uv ∈ E and (S \ {v}) ∪{u} is a dominating set. The minimum cardinality of a secure dominating set in G is the secure domination number 𝛾𝑠 (G). We present few bounds on these parameter band certain graphs for which equality of both parameter holds

Keywords

Co-secure domination, Securedomination, Jahangir graphs, Friendship graphs, Helm graph.

References

[1] A. P. Burger, M. A. Henning and J. H. van Vuuren, “Vertex covers and secure domination in graphs”, Quaest. Math.,vol.31,pp.163-171, 2008.
[2] D.A.Mojdeh and A. N.Ghameshlou, “Domination in Jahangir Graph, J2m”, Int.J.Contemp.Math. Sciences, vol.2, pp.1193-1199, 2007.
[3] E. J. Cockayne, P. J.P. Grobler,et.at “Protection of a graph”, Util. Math., vol.67, pp.19-32, 2005.
[4] S. Arumugan,KaramEbadi and Martin Manrique “Co-Secure and Secure Domination in Graphs”, Util. Math., vol.94, pp.167-182, 2014.
[5] Zepeng Li, Zehui Shao and Jin Xu “On secure domination in trees”, Quaest. Math., vol.40, pp.1-12, 2017.