2-connected graphs with small 2-connected dominating sets

Let G be a 2-connected graph. A subset D of V(G) is a 2-connected dominating set if every vertex of G has a neighbor in D and D induces a 2-connected subgraph. Let γ2(G) denote the minimum size of a 2-connected dominating set of G. Let δ(G) be the minimum degree of G. For an n-vertex graph G, we prove thatγ2(G)⩽n ln δ(G)δ(G) (1+oδ(1))where oδ(1) denotes a function that tends to 0 as δ→∞. The upper bound is asymptotically tight. This extends the results in (Arnautov, Prikl. Mat. i Programmirovanie 11 (1974) 3–8, Caro et al., SIAM J. Discrete Math. 13 (2000) 202–211, Lovász, Discrete Math. 13 (1975) 383–390 and Payan, Cahièrs Centre Etudes Rech. Opér. 17 (1975) 307–317).