Some remarks on α-domination

Franz Dahme; Dieter Rautenbach; Lutz Volkmann

Discussiones Mathematicae Graph Theory (2004)

  • Volume: 24, Issue: 3, page 423-430
  • ISSN: 2083-5892

Abstract

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Let α ∈ (0,1) and let G = ( V G , E G ) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set D ⊆ V G is called an α-dominating set of G, if | N G ( u ) ∩ D | ≥ α d G ( u ) for all u ∈ V G ∖ 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.

How to cite

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Franz Dahme, Dieter Rautenbach, and Lutz Volkmann. "Some remarks on α-domination." Discussiones Mathematicae Graph Theory 24.3 (2004): 423-430. <http://eudml.org/doc/270246>.

@article{FranzDahme2004,
abstract = {Let α ∈ (0,1) and let $G = (V_G,E_G$) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D ⊆ V_G$ is called an α-dominating set of G, if $|N_G(u) ∩ D| ≥ αd_G(u)$ for all $u ∈ V_G∖D$. 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.},
author = {Franz Dahme, Dieter Rautenbach, Lutz Volkmann},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {α-domination; domination},
language = {eng},
number = {3},
pages = {423-430},
title = {Some remarks on α-domination},
url = {http://eudml.org/doc/270246},
volume = {24},
year = {2004},
}

TY - JOUR
AU - Franz Dahme
AU - Dieter Rautenbach
AU - Lutz Volkmann
TI - Some remarks on α-domination
JO - Discussiones Mathematicae Graph Theory
PY - 2004
VL - 24
IS - 3
SP - 423
EP - 430
AB - Let α ∈ (0,1) and let $G = (V_G,E_G$) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set $D ⊆ V_G$ is called an α-dominating set of G, if $|N_G(u) ∩ D| ≥ αd_G(u)$ for all $u ∈ V_G∖D$. 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.
LA - eng
KW - α-domination; domination
UR - http://eudml.org/doc/270246
ER -

References

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  1. [1] N. Alon and J. Spencer, The probabilistic method, 2nd ed., (Wiley-Interscience Series in Discrete Math. and Optimization, 2000), doi: 10.1002/0471722154. Zbl0996.05001
  2. [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. Zbl0048.11804
  3. [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. [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. Zbl0602.05043
  5. [5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of domination in graphs (Marcel Dekker, New York, 1998). Zbl0890.05002
  6. [6] F. Dahme, D. Rautenbach and L. Volkmann, α-domination perfect trees, manuscript (2002). 
  7. [7] O. Ore, Theory of Graphs, Amer. Math. Soc. Colloq. Publ., 38 (Amer. Math. Soc., Providence, RI, 1962). 
  8. [8] C. Payan and N.H. Xuong, Domination-balanced graphs, J. Graph Theory 6 (1982) 23-32, doi: 10.1002/jgt.3190060104. Zbl0489.05049
  9. [9] D.R. Woodall, Improper colourings of graphs, Pitman Res. Notes Math. Ser. 218 (1988) 45-63. 

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