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 ${\displaystyle {\gamma }_{k,p}\left(G\right)}$ denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that ${\displaystyle {\gamma }_{k,p}\left(G\right)\le \left(\frac{p}{p+k}\right)n\left(G\right)}$ 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 ${\displaystyle {\gamma }_{2,2}\left(G\right)\le \frac{n\left(G\right)+1}{2}}$ for all connected graphs G and characterize all connected graphs with ${\displaystyle {\gamma }_{2,2}=\frac{n+1}{2}}$. This means that for k = p = 2 we characterize all connected graphs for which the conjecture is true without the precondition that δ₂ ≥ 3.