Approximating k-Connected m-Dominating Sets

A subset S of nodes in a graph G is k-connected m-dominating set ((k , m)-cds) if the subgraph G[S] induced by and every $$v \in V {\setminus } S$$ has at least m neighbors S. In k -Connected -Dominating Set m)-CDS) problem, goal to find minimum weight (k, m)-cds node-weighted graph. For $$m \ge k$$ we obtain following approximation ratios. unit disk graphs improve ratio $$O(k \ln k)$$ Nutov (Inf Process Lett 140:30–33, 2018) $$\min \left\{ \frac{m^2}{(m-k+1)^2},k^{2/3}\right\} \cdot O(\ln ^2 —this first sublinear for polylogarithmic $$O(\ln k)/\epsilon ^2$$ when (1+\epsilon )k$$ ; furthermore, \frac{m}{m-k+1},\sqrt{k}\right\} uniform weights. general our n)$$ improves previous best $$O(k^2 (2018) matches known weights Zhang et al. (INFORMS J Comput 30(2):217–224, 2018). These results are obtained showing same ratios Subset -Connectivity problem terminals an set.