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Degree Equitable Domination Number and Independent Domination Number of a Graph

A.Nellai Murugan1, G.Victor Emmanuel2
  1. Assoc. Prof. of Mathematics, V.O. Chidambaram College, Thuthukudi-628 008, Tamilnadu, India
  2. Asst. Prof. of Mathematics, St. Mother Theresa Engineering College, Thuthukudi-628 102, Tamilnadu, India
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Abstract

In this paper, we introduce the degree equitable domination number in graphs. Some interesting relationships are identified between domination and degree equitable domination and independent domination. It is also shown that any positive integers with are realizable as the domination number, degree equitable domination number and independent domination number of a graph. We also investigate the degree equitable domination in the corona of graphs.

Keywords

Degree equitable dominating set, degree equitable domination number,

I. INTRODUCTION

The concept of domination in graphs evolved from a chess board problem known as the Queen problem- to find the minimum number of queens needed on an 8x8 chess board such that each square is either occupied or attacked by a queen. C.Berge[3] in 1958 and 1962 and O.Ore[8] in 1962 started the formal study on the theory of dominating sets. Thereafter several studies have been dedicated in obtaining variations of the concept. The authors in [7] listed over 1200 papers related to domination in graphs in over 75 variation.
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References

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[2] I.S. Aniversario, F.P. Jamil and S.R. Canoy Jr., The closed geodetic numbers of graphs, Utilitas Mathematica, Vol. 74, pp. 3-18, 2007.

[3] C. Berge, theory of Graphs and its Applications, Methuen, London, 1962.

[4] F. Buckley, F. Harary. Distance in graphs. Redwood City. C. A: Addition-Wesley. 1990.

[5] W. Duckworth and N. C. Wormald, On the independent domination number of random regular graphs, Combinatorics, Probabilty and Computing, Vol. 15, 4, 2006.

[6] T. Haynes, S. Hedetniemi and M. Henning, Domination in graphs applied to electrical power networks, J. Discrete Math. 15(4), 519-529, 2000.

[7] T.W. Hanes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs. Marcel Dekker, Inc. New York (1998).

[8] O. Ore, Theory of graphs, Amer. Math. Soc. Colloq. Publ., Vol.38, Providence, 1962.

[9] L.Sun and J. Wang, An upper bound for the independent domination number, Journal of Combinatorial Theory, Vol.76, 2. 240-246, 1999.

[10] H. Walikar, B. Acharya and E. Sampathkumar, Recent developments in the theory of domination in graphs, Allahabad, 1, 1979.

[11] Teresa L. Tacabobo and Ferdinand P.Jamil, Closed Domination in Graphs, International Mathematical Forum, Vol. 7, 2012, No. 51, 2509-2518.

[12] T.R.Nirmala Vasantha, A study on Restrained Domination number of a graph, Thesis 2007, M.S.U, Tirunelveli.

[13] B.D.Acharya, H.B.Walikar, and E.Sampathkumar, Recent developments in the theory of domination in graphs. In MRI Lecture Notes in Math. Mehta Research Instit, Allahabad No.1, (1979).