Adaptive power topology control (APTC) is a local algorithm for constructing a one-parameter family of thetas-graphs, where each node increases power until it has a neighbor in every thetas sector around it. We show it is possible to use such a local geometric thetas-constraint to ensure full network connectivity, and consider tradeoffs between assumptions about the wireless footprint and constraints on the boundary nodes. In particular, we show that if the boundary nodes can communicate with neighboring boundary nodes and all interior nodes satisfy a thetas
I<pi constraint, we can guarantee connectivity for any arbitrary wireless footprint. If we relax the boundary assumption and instead impose a thetas
B<3pi/2 constraint on the boundary nodes, together with the thetas
I<pi constraint on interior nodes, we can guarantee full network connectivity using only a "weak-monotonicity" footprint assumption. The weak-monotonicity model, introduced herein, is much less restrictive than the disk model of coverage and captures aspects of the spatial correlations inherent in signal propagation and noise. We show that under the idealized disk model of coverage, APTC constructs graphs that are sparse. Finally, we show that if the wireless footprint has sufficiently small "eccentricity", then there is some thetas for which greedy geometric routing always succeeds
