Some remarks on α-domination

ArticleOriginal scientific text

Title

Authors ¹, ², ¹

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 \subseteq V_{G}

is called an α-dominating set of G, if

| N_{G} (u) \cap D | \geq α d_{G} (u)

for all

u \in 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.

α-domination, domination

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