Adaptive power topology control (APTC) is a local algorithm for constructing a one-parameter family of θ-graphs, where each node increases power until it has a neighbor in every θ sector around it. We show it is possible to use such a local geometric θ-constraint to determine whether full network connectivity is achievable, and consider tradeoffs between assumptions of the wireless footprint and constraints on the boundary nodes. In particular, we show that if the boundary nodes can communicate

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... with neighboring boundary nodes and all interior nodes satisfy a θ I < π constraint, we can guarantee connectivity for any arbitrary wireless footprint. If we relax the boundary assumption and instead impose a θ B < 3π/2 constraint on the boundary nodes, together with the θ I < π constraint on interior nodes, we can guarantee full network connectivity using only a "weak-monotonicity" footprint assumption. The weakmonotonicity 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. Finally, assuming the idealized disk model of coverage, we show that when θ < π, APTC constructs graphs that are sparse, and when θ < 2π/3, the graphs support greedy geometric routing.