DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

Journal Impact Factor (JIF 2022): 0.7

5-year Journal Impact Factor (2022): 0.7

CiteScore (2022): 1.9

SNIP (2022): 0.902

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 24(3) (2004) 423-430
DOI: https://doi.org/10.7151/dmgt.1241

SOME REMARKS ON α-DOMINATION

Franz Dahme

Lehrstuhl II für Mathematik, RWTH-Aachen
52056 Aachen, Germany

e-mail: franz.dahme@web.de

Dieter Rautenbach

Forschungsinstitut für Diskrete Mathematik
Lennéstr. 2, D-53113 Bonn, Germany

e-mail: rauten@or.uni-bonn.de

Lutz Volkmann

Lehrstuhl II für Mathematik, RWTH-Aachen
52056 Aachen, Germany

e-mail: volkm@math2.rwth-aachen.de

Abstract

Let α ∈ (0,1) and let G = (VG,EG) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set D ⊆ VG is called an α-dominating set of G, if |NG(u)∩D| ≥ αdG(u) for all u ∈ VG∖D. We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

Keywords: α-domination; domination.

2000 Mathematics Subject Classification: 05C69.

References

[1] N. Alon and J. Spencer, The probabilistic method, 2nd ed., (Wiley-Interscience Series in Discrete Math. and Optimization, 2000), doi: 10.1002/0471722154.
[2] H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations, Ann. Math. Stat. 23(1952) 493-507, doi: 10.1214/aoms/1177729330.
[3] J.E. Dunbar, D.G. Hoffman, R.C. Laskar and L.R. Markus, α-domination, Discrete Math. 211 (2000) 11-26, doi: 10.1016/S0012-365X(99)00131-4.
[4] J.F. Fink, M.S. Jacobson, L.F. Kinch and J. Roberts, On graphs having domination number half their order, Period. Math. Hungar. 16 (1985) 287-293, doi: 10.1007/BF01848079.
[5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of domination in graphs (Marcel Dekker, New York, 1998).
[6] F. Dahme, D. Rautenbach and L. Volkmann, α-domination perfect trees, manuscript (2002).
[7] O. Ore, Theory of Graphs, Amer. Math. Soc. Colloq. Publ., 38 (Amer. Math. Soc., Providence, RI, 1962).
[8] C. Payan and N.H. Xuong, Domination-balanced graphs, J. Graph Theory 6 (1982) 23-32, doi: 10.1002/jgt.3190060104.
[9] D.R. Woodall, Improper colourings of graphs, Pitman Res. Notes Math. Ser. 218 (1988) 45-63.

Received 31 March 2003
Revised 12 December 2003


Close