In this paper a novel power control scheme for establishing end-to-end connectivity over Wireless Ad-Hoc and Sensor Networks (WASN) is presented. Based on a deterministic connectivity model, the proposed scheme exploits overlapping information related to the properties of the network's Delaunay graph and localization. Sufficient conditions for establishing end-to-end connectivity over the WASN are provided; these are strongly attached to the properties of the network's Delaunay graph, formed by the nodes'spatial distribution. Under the provision of the Delaunay-related overlapping information to the network's operating nodes, these sufficient conditions are utilized for the construction of a distributed Shortest Path-like algorithm. Without any additional network interaction, the resulting Distributed Delaunay Connectivity Algorithm (DDelCA) is capable of adjusting the nodal transmission power to the minimum value that establishes the network's end-to-end connectivity. Extensive simulation results are offered to evaluate the network's performance using the DDelCA transmission power configuration and highlight the benefits of the DDelCA scheme.