Trees with equal 2-domination and 2-independence numbers

Abstract

Let G = (V, E) be a graph. A subset S of V is a 2-dominating set if every vertex of V - S is dominated at least 2 times, and S is a 2-independent set of G if every vertex of S has at most one neighbor in S. The minimum cardinality of a 2-dominating set a of G is the 2-domination number γ2(G) and the maximum cardinality of a 2-independent set of G is the 2-independence number β2(G). Fink and Jacobson proved that γ2(G) ≤ β2(G) for every graph G. In this paper we provide a constructive characterization of trees with equal 2-domination and 2-independence numbers.