Minimum 2-dominating sets in regular graphs

Abstract

Given a graph G , a set D k ⊆ V ( G ) is said to be k -dominating if every vertex in V ( G ) ∖ D k has at least k neighbors in D k . The k -domination number γ k ( G ) is the smallest possible cardinality of such a set D k . In this work, for any fixed d ≥ 3 , we obtain upper bounds, valid asymptotically almost surely for random d -regular graphs, on γ 2 ( G ) / | V ( G ) | as a function of d . In addition to this, we show that essentially the same bounds hold for all d -regular graphs G with sufficiently large girth.