Discussiones Mathematicae Graph Theory 25(1-2) (2005)
29-34
DOI: https://doi.org/10.7151/dmgt.1256
ON DOUBLE DOMINATION IN GRAPHS
Jochen Harant
Department of Mathematics |
Michael A. Henning
School of Mathematics, Statistics, & |
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
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Received 22 October 2003
Revised 6 May 2004
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