Consider a placement of heterogeneous, wireless sensors that can vary the transmission range by increasing or decreasing power. The problem of determining an optimal assignment of transmission radii, so that the resulting network is strongly-connected and more generally k-connected has been studied in the literature. In traditional k-connectedness, the network is able resist the failure of up to k . 1 nodes anywhere in the network, and still remain strongly-connected. In this paper we introduce a much stronger k-connectedness property, which we show can be implemented efficiently, and without great increase in the radii of transmission needed to simply achieve connectedness. We say that a network is dominating kconnected if, for any simultaneous failure of nodes throughout the network, with at most k . 1 nodes failures occurring in the out-neighborhood any surviving (up) node, the set U of up nodes forms a dominating set and induces a strongly-connected subdigraph. In this paper, we give a simple characterization of the networks that are dominating k-connected and design an associated efficient algorithm for determining the dominating connectivity, i.e., the maximum k such that the network is dominating k-connected. We also present an efficient algorithm for computing an assignment of transmission radii that results in a dominating k-connected network which minimizes the maximum radius. Furthermore, we show that the maximum radius in this assignment is no more than a multiplicative factor of k greater than the percolation radius ?perc, i.e., the minimum that the maximum transmission radius can be so that the network remains connected. We show through empirical testing that this multiplicative factor can, in practice, be considerably less than k and only slightly greater than that required to achieve traditional k-connectedness. Finally, we show that for sensors placed on the lattice points of a two-dimensional square, we can achieve dominating k-connectedness with a multiplicative factor of at most