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 ∆( G ) ≥ k ≥ 2, γ k ( G ) ≥ γ ( G ) + k − 2, and then γ k ( G ) > γ ( G ) for any k ≥ 3, where γ ( G ) = γ 1 ( G ) is the usual domination number. Thus, it is the most interesting problem to characterize graphs G with γ 2 ( G ) = γ ( G ). In this paper, we characterize outerplanar graphs with equal 2-domination and domination numbers.