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 that γ2(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 [3,5,10,12].