Discussiones Mathematicae Graph Theory 25(1-2) (2005) 7-12
doi: 10.7151/dmgt.1254

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Frank Göring

Department of Mathematics
Chemnitz University of Technology
D-09107 Chemnitz, Germany
e-mail: frank.goering@mathematik.tu-chemnitz.de

Jochen Harant

Department of Mathematics
Technical University of Ilmenau
D-98684 Ilmenau, Germany
e-mail: harant@mathematik.tu-ilmenau.de


For a finite undirected graph G on n vertices two continuous optimization problems taken over the n-dimensional cube are presented and it is proved that their optimum values equal the domination number γ of G. An efficient approximation method is developed and known upper bounds on γ are slightly improved.

Keywords: graph, domination.

2000 Mathematics Subject Classification: 05C69.


[1] Y. Caro, New results on the independence number (Technical Report, Tel-Aviv University, 1979).
[2] Y. Caro and Zs. Tuza, Improved lower bounds on k-independence, J. Graph Theory 15 (1991) 99-107, doi: 10.1002/jgt.3190150110.
[3] R. Diestel, Graph Theory, Graduate Texts in Mathematics (Springer, 1997).
[4] N. Alon, J.H. Spencer and P. Erdös, The Probabilistic Method (John Wiley and Sons, Inc. 1992), page 6.
[5] M.R. Garey and D.S. Johnson, Computers and Intractability, A Guide to the Theory of NP-Completeness (W.H. Freeman and Company, San Francisco, 1979).
[6] J. Harant, Some news about the independence number of a graph, Discuss. Math. Graph Theory 20 (2000) 71-79, doi: 10.7151/dmgt.1107.
[7] J. Harant, A. Pruchnewski and M. Voigt, On dominating sets and independent sets of graphs, Combinatorics, Probability and Computing 8 (1999) 547-553, doi: 10.1017/S0963548399004034.
[8] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, Inc., New York, N.Y., 1998), page 52.
[9] V.K. Wei, A lower bound on the stability number of a simple graph (Bell Laboratories Technical Memorandum 81-11217-9, Murray Hill, NJ, 1981).

Received 23 September 2003
Revised 15 June 2004