Discussiones Mathematicae Graph Theory 25(1-2) (2005) 29-34
doi: 10.7151/dmgt.1256

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ON DOUBLE DOMINATION IN GRAPHS

Jochen Harant

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

Michael A. Henning

School of Mathematics, Statistics, &
Information Technology, University of KwaZulu-Natal
Pietermaritzburg, 3209 South Africa
e-mail: henning@ukzn.ac.za

Abstract

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ ×2(G). A function f(p) is defined, and it is shown that γ ×2(G) = minf(p), where the minimum is taken over the n-dimensional cube Cn = {p = (p1,…,pn) | pi ∈ IR, 0 ≤ pi ≤ 1,i = 1,…,n}. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then γ×2(G) ≤ ((ln(1+d)+lnδ+1)/δ)n.

Keywords: average degree, bounds, double domination, probabilistic method.

2000 Mathematics Subject Classification: 05C69.

References

[1] M. Blidia, M. Chellali and T.W. Haynes, Characterizations of trees with equal paired and double domination numbers, submitted for publication.
[2] M. Blidia, M. Chellali, T.W. Haynes and M.A. Henning, Independent and double domination in trees, submitted for publication.
[3] M. Chellali and T.W. Haynes, Paired and double domination in graphs, Utilitas Math., to appear.
[4] J. Harant, A. Pruchnewski and M. Voigt, On dominating sets and independent sets of graphs, Combin. Prob. and Comput. 8 (1998) 547-553, doi: 10.1017/S0963548399004034.
[5] F. Harary and T.W. Haynes, Double domination in graphs, Ars Combin. 55 (2000) 201-213.
[6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
[7] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998).
[8] M.A. Henning, Graphs with large double domination numbers, submitted for publication.
[9] C.S. Liao and G.J. Chang, Algorithmic aspects of k-tuple domination in graphs, Taiwanese J. Math. 6 (2002) 415-420.
[10] C.S. Liao and G.J. Chang, k-tuple domination in graphs, Information Processing Letters 87 (2003) 45-50, doi: 10.1016/S0020-0190(03)00233-3.

Received 22 October 2003
Revised 6 May 2004