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 2 V (G) n D is within distance k to at least p vertices in D. The parameter 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 k;p(G) p p+k n(G) for any graph G with k(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 conrmed this conjecture for all integers k and p for the case that p is a multiple of k. In this paper we show that 2;2(G) (n(G) + 1)=2 for all connected graphs G and characterize all connected graphs with 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 2 3.