Geodesic
On double domination in graphs
Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 1-2, pp. 29-34.

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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) = min f(p), where the minimum is taken over the n-dimensional cube Cⁿ = p = (p₁,...,pₙ) | p_i ∈ IR, 0 ≤ p_i ≤ 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 
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Harant, Jochen; Henning, Michael. On double domination in graphs. Discussiones Mathematicae. Graph Theory, Tome 25 (2005) no. 1-2, pp. 29-34. http://geodesic.mathdoc.fr/item/DMGT_2005_25_1-2_a2/

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