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

摘要

A subset D of the vertex set of a graph O 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, Helming, and Swart posed the conjecture that γ_(k,p)(G) ≤ p/(p+k)n(G) for any graph G with δ_k(G) ≥ k + p - 1 which means that every vertex is within distance fc 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.