  We investigate the condition on transmission radius needed to achieve
connectivity in duty-cycled wireless sensor networks (briefly, DC-WSN). First,
we settle a conjecture of Das et. al. (2012) and prove that the connectivity
condition on Random Geometric Graphs (RGG), given by Gupta and Kumar (1989),
can be used to derive a weak sufficient condition to achieve connectivity in
DC-WSN. To find a stronger result, we define a new vertex-based random
connection model which is of independent interest. Following a proof technique
of Penrose (1991) we prove that when the density of the nodes approaches
infinity then a finite component of size greater than 1 exists with probability
0 in this model. We use this result to obtain an optimal condition on node
transmission radius which is both necessary and sufficient to achieve
connectivity and is hence optimal. The optimality of such a radius is also
tested via simulation for two specific duty-cycle schemes, called the
contiguous and the random selection duty-cycle scheme. Finally, we design a
minimum-radius duty-cycling scheme that achieves connectivity with a
transmission radius arbitrarily close to the one required in Random Geometric
Graphs. The overhead in this case is that we have to spend some time computing
the schedule.
