Robust sensor range for constructing strongly connected spanning digraphs in UDGs

We study the following problem: Given a set of points in the plane and a positive integer k > 0, construct a geometric strongly connected spanning digraph of out-degree at most k and whose longest edge length is the shortest possible. The motivation comes from the problem of replacing omnidirectional antennae in a sensor network with k directional antennae per sensor so that the resulting sensor network is strongly connected. The contribution of this is paper is twofold: 1) We introduce a notion of robustness of the radius in geometric graphs. This allows us to provide stronger lower bounds for the edge length needed to solve our problem, while nicely connecting two previously unrelated research directions (graph toughness and multiple directional antennae). 2) We present novel upper bound techniques which, in combination with stronger lower bounds, allow us to improve the previous approximation results for the edge length needed to achieve strong connectivity for k = 4 (from 2sin(π/5) to optimal) and k = 3 (from 2 sin(π/4) to 2 sin(2π/9))