INTEGRITY AND DOMINATION INTEGRITY OF GEAR GRAPHS

Yıl 2016, Cilt: 6 Sayı: 1, 54 - 63, 01.06.2016

- R.sundareswaran - V.swaminathan

Öz

C.A. Barefoot, et. al. [4] introduced the concept of the integrity of a graph. It is an useful measure of vulnerability and it is defined as follows. I G = min{|S| + m G − S : S ⊂ V G }, where m G − S denotes the order of the largest component in G − S. Unlike the connectivity measures, integrity shows not only the difficulty to break down the network but also the damage that has been caused. A subset S of V G is said to be an I-set if I G = |S| + m G − S . We introduced a new vulnerability parameter in [4],namely domination integrity of a graph G. It is a defined as DI G = min{|S| + m G − S }, where S is a dominating set of G and m G − S denotes the order of the largest component in G − S. K.S. Bagga,et. al. [2] gave a formula for I K2 × Cn . In this paper, we give a correct formula for I K2 × Cn . We find some results on the integrity and domination integrity of gear graphs

Anahtar Kelimeler

Connectivity, Network Design and Communication, vulnerability, Integrity , Domination Integrity, Gear Graph.

Kaynakça

• K.S.Bagga, L.W. Beineke, W.D. Goddard, M.J. Lipman and R.E.Pippert, A surety of Integrity, Dis- crete Applied Mathematics, 37/38, (1992), 13-28.
• K.S. Bagga, L.w. Beineke and R.E. Pippert, The integrity of Prisms (Preliminary report), Abstracts Amer. Math. Soc. 10, 12, 1989.
• J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, Macmillan, London and Elsevier, Newyork , 1976.
• C.A. Barefoot, R. Entringer and Henda C. Swart,Vulnerability in graphs A comparative survey, J. Combin. Math. Combin. Comput. 1 (1987),1322.
• A. Brandstadt, V. B. Le and J. P. Spinrad, Graph classes: a survey, SIAM, Philadelphia, PA, 1999. [6] J. A. Bondy, and U. S. R. Murty, Graph Theory with Applications, The McMilan Press Ltd., 1976.
• M. Cozzens, D. Moazzami, S. Stueckle, The tenacity of a graph, Proc. Seventh International Conf. on the Theory and Application of Graphs, Wiley, New York, 1995, pp. 1111-1122.
• P.D.Chawathe and S.A.Shede , Integrity of P2× Pn, Number Theory and Discrete Mathematics, Trends in Mathematics 2002, pp 149-155.
• Ersin Aslan and Alpay Kirlangic, Computing The Scattering Number and The Toughness for Gear Graphs, Bulletin of International Mathematical Virtual Institute, ISSN 1840-4367 , Vol. 1(2011), 1-11. [10] Fengwei Li, Some Results on Tenacity of Graphs, WSEAS Transcations on Mathematics, Issue 9, Volume 11, September 2012, 760 -772.
• Goddard, W., Swart, H.C.,Integrity in graphs : Bounds and Basics, J. Combin. Math. Combin. Comput. 7(1990), 139-151.
• H.A. Jung, On a class of posets and the corresponding comparability graphs, J. Combin. Theory Ser. B 24 (1978) 125-133.
• A. Kirlangic, The Rupture Degree and Gear Graphs, Bull. Malays. Math. Sci. Soc., Vol.32, 2009, (1), pp. 31-36.
• Y. K. Li , S. G. Zhang and X. L. Li, Rupture degree of graphs, Int. J. Comput. Math., Vo. 82, 2005 (7), 793 - 803.
• R. Sundareswaran and V. Swaminathan, Domination Integrity in Graphs, Proceedings of International Conference on Mathematical and Experimental Physics, Prague, 3-8 August 2009, pp. 46-57.
• R. Sundareswaran and V. Swaminathan, Domination Integrity of Middle Graphs, Algebra, Graph Theory and Their Applications, T. Chelvam, S. Somasundaram and R. Kala, Eds.,, Narosa Publishing House, New Delhi, 2010, pp. 88-92.
• R. Sundareswaran and V. Swaminathan, Domination Integrity in Trees, Bulletin of International Mathematical Virtual Institute,ISSN 1840-4367, Vol. 2 (2012), 153-161.
• R. Sundareswaran and V. Swaminathan, Domination Integrity of Powers of Cycles, International Journal of Mathematical Research, 3(3),2011, 257-265.
• R.Sundareswaran for the photography and short autobiography, see TWMS J. App. Eng. Math., V.5, N.1.
• V.Swaminathan for the photography and short autobiography, see TWMS J. App. Eng. Math., V.5, N.1.