Geodesic
A remark on the (2,2)-domination number
Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 2, pp. 361-366

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A subset D of the vertex set of a graph G is a (k,p)-dominating set if every vertex v ∈ V(G)∖D is within distance k to at least p vertices in D. The parameter γ_k,p(G) denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that γ_k,p(G) ≤ (p/(p+k))n(G) for any graph G with δₖ(G) ≥ k+p-1, where the latter means that every vertex is within distance k to at least k+p-1 vertices other than itself. In 2005, Fischermann and Volkmann confirmed this conjecture for all integers k and p for the case that p is a multiple of k. In this paper we show that γ_2,2(G) ≤ (n(G)+1)/2 for all connected graphs G and characterize all connected graphs with γ_2,2 = (n+1)/2. This means that for k = p = 2 we characterize all connected graphs for which the conjecture is true without the precondition that δ₂ ≥ 3.
Keywords: domination, distance domination number, p-domination number 
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Korneffel, Torsten; Meierling, Dirk; Volkmann, Lutz. A remark on the (2,2)-domination number. Discussiones Mathematicae. Graph Theory, Tome 28 (2008) no. 2, pp. 361-366. http://geodesic.mathdoc.fr/item/DMGT_2008_28_2_a10/