## Abstract

Let *G* = (*V,E*) be a graph and *S* ⊆ *V*. We say that *S* is a dominating set of *G*, if each vertex in *V* \ *S* has a neighbor in *S*. Moreover, we say that *S* is a connected (respectively, 2-edge connected or 2-connected) dominating set of *G* if *G*[*S*] is connected (respectively, 2-edge connected or 2-connected). The domination (respectively, connected domination, or 2- edge connected domination, or 2-connected domination) number of *G* is the cardinality of a minimum dominating (respectively, connected dominating, or 2-edge connected dominating, or 2-connected dominating) set of *G*, and is denoted γ(*G*) (respectively γ_{1}(*G*), or γ′ _{2}(*G*), or γ_{2}(*G*)). A well-known result of Duchet and Meyniel states that γ_{1}(*G*) ≤ 3γ(*G*) − 2 for any connected graph *G*. We show that if γ(*G*) ≥ 2, then γ′ _{2}(*G*) ≤ 5γ(*G*) − 4 when *G* is a 2-edge connected graph and γ_{2}(*G*) ≤ 11γ(*G*) − 13 when *G* is a 2-connected triangle-free graph.