Advanced Search

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 ${\gamma }_{k,p}\left(G\right)$ denotes the minimum cardinality of a (k,p)-dominating set of G. In 1994, Bean, Henning and Swart posed the conjecture that ${\gamma }_{k,p}\left(G\right)\le (p/(p+k))n\left(G\right)$ 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...