The aim of topology control was determining an appropriate topology for wireless ad hoc networks. In wireless ad hoc networks geometric routing protocol was a kind of important routing protocols

which requires the underlying topology be planar

sparse

connected and t-spanner in order to guarantee the delivery of message and bound the routing length. A new geometry structure named AUDel was proposed and two efficient algorithms to construct the new structure were proposed. Theoretic analysis showed that AUDel satisfies those properties

and our algorithms are much more communication efficient than previous algorithms which construct planar t-spanner of the original topology. These conclusions are confirmed by simulation results.