PDF
Discussiones Mathematicae Graph Theory 30(2) (2010)
277-288
DOI: https://doi.org/10.7151/dmgt.1494
RANDOM PROCEDURES FOR DOMINATING SETS IN BIPARTITE GRAPHS
Sarah Artmann and Jochen Harant
Institut für Mathematik, TU Ilmenau
Postfach 100565, D-98684 Ilmenau, Germany
e-mails: {sarah.artmann,jochen.harant}@tu-ilmenau.de
Abstract
Using multilinear functions and random procedures, new upper bounds on the domination number of a bipartite graph in terms of the cardinalities and the minimum degrees of the two colour classes are established.Keywords: domination, bipartite graph, multilinear function, random procedure.
2010 Mathematics Subject Classification: 05C69.
References
[1] | N. Alon and J. Spencer, The Probabilistic Method (John Wiley and Sons, Inc., 1992). |
[2] | V.I. Arnautov, Estimation of the exterior stability number of a graph by means of the minimal degree of the vertices, (Russian), Prikl. Mat. Programm. 11 (1974) 3-8. |
[3] | S. Artmann, F. Göring, J. Harant, D. Rautenbach and I. Schiermeyer, Random procedures for dominating sets in graphs, submitted. |
[4] | Y. Caro, New results on the independence number (Technical Report. Tel-Aviv University, 1979). |
[5] | Y. Caro and Y. Roditty, On the vertex-independence number and star decomposition of graphs, Ars Combin. 20 (1985) 167-180. |
[6] | Y. Caro and Y. Roditty, A note on the k-domination number of a graph, Internat. J. Math. Sci. 13 (1990) 205-206, doi: 10.1155/S016117129000031X. |
[7] | G.J. Chang and G.L. Nemhauser, The k-domination and k-stability problems in sun-free chordal graphs, SIAM J. Algebraic Discrete Methods 5 (1984) 332-345, doi: 10.1137/0605034. |
[8] | F. Göring and J. Harant, On domination in graphs, Discuss. Math. Graph Theory 25 (2005) 7-12, doi: 10.7151/dmgt.1254. |
[9] | J. Harant and A. Pruchnewski, A note on the domination number of a bipartite graph, Ann. Combin. 5 (2001) 175-178, doi: 10.1007/PL00001298. |
[10] | J. Harant, A. Pruchnewski, and M. Voigt, On dominating sets and independendent sets of graphs, Combin. Prob. Comput. 8 (1999) 547-553, doi: 10.1017/S0963548399004034. |
[11] | J. Harant and D. Rautenbach, Domination in bipartite graphs, Discrete Math. 309 (2009) 113-122, doi: 10.1016/j.disc.2007.12.051. |
[12] | T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of domination in graphs (Marcel Dekker, Inc., New York, 1998). |
[13] | T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Domination in Graphs Advanced Topics (Marcel Dekker, Inc., New York, 1998). |
[14] | C. Payan, Sur le nombre d'absorption d'un graphe simple, (French), Cah. Cent. Étud. Rech. Opér. 17 (1975) 307-317. |
[15] | 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 12 November 2008
Revised 10 August 2009
Accepted 9 November 2009
Close