2-connected graphs with small 2-connected dominating sets.
The main result of the paper is the following. Let \(G\) be a \(2\)-connected graph with \(n\) vertices and minimum degree \(\delta\). Then \(\gamma_2 \leq n(\ln(\delta)/\delta)(1+o_{\delta}(1))\), where \(\gamma_2\) is the minimum size of a \(2\)-connected dominating set of \(G\) and \(o_{\delta}(1)\) denotes a function that tends to 0 as \(\delta \rightarrow \infty\). The proof uses the probabilistic method. This result extends similar results for \(\gamma_1\), the size of a minimum connected dominating set [see the authors and \textit{D. West}, SIAM J. Discrete Math. 13, No. 2, 202--211 (1998; Zbl 0941.05045), \textit{L. Lovász}, Discrete Math. 13, 383--390 (1975; Zbl 0323.05127)].
scientific article; zbMATH DE number 1967276