Characterization of outerplanar graphs with equal 2-domination and domination numbers

A k-domination number of a graph G is minimum cardinality of a k-dominating set of G, where a subset S ⊆ V(G) is a k-dominating set if each vertex v ∈ V(G) \ S is adjacent to at least k vertices in S. It is known that for any graph G with Δ(G) ≥ k ≥ 2, γk(G) ≥ γ(G) + k – 2, and then γk(G) \u3e γ(G) for any k ≥ 3, where γ(G) = γ1(G) is the usual domination number. Thus, it is the most interesting problem to characterize graphs G with γ2(G) = γ(G). In this paper, we characterize outerplanar graphs with equal 2-domination and domination numbers