A remark on the (2, 2)-domination number

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 γk,p(G) denotes the minimum cardinality of a (k, p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that γk,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 γ2,2(G) ≤ (n(G)+1)/2 for all connected graphs G and characterize all connected graphs with γ2,2 = (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.