A remark on a conjecture for the (k, p)-domination number

Abstract

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+kn(G) for any graph G with δ_k (G) ≥ k + p - 1 which means that every vertex is within distance k to at least k + p - 1 vertices other than itself. In this note we confirm this conjecture for all integers k and p when p is a multiple of k. In the remaining cases we present some weaker statements.