### abstract

- We study a combinatorial geometric problem related to the design of wireless networks with directional antennas. Specifically, we are interested in necessary and sufficient conditions on such antennas that enable one to build a connected communication network, and in efficient algorithms for building such networks when possible. We formulate the problem by a set of n points in the plane, indicating the positions of n transceivers. Each point is equipped with an α-degree directional antenna, and one needs to adjust the antennas (represented as wedges), by specifying their directions, so that the resulting (undirected) communication graph G is connected. (Two points are connected by an edge in G, if and only if q lies in pʼs wedge and p lies in qʼs wedge.) We prove that if , then it is always possible to adjust the wedges so that G is connected, and that is sometimes necessary to achieve this. Our proof is constructive and yields an time algorithm for adjusting the wedges, where k is the size of the convex hull of . Sometimes it is desirable that the communication graph G contain a Hamiltonian path. By a result of Fekete and Woeginger (1997) [8], if , then it is always possible to adjust the wedges so that G contains a Hamiltonian path. We give an alternative proof to this, which is interesting, since it produces paths of a different nature than those produced by the construction of Fekete and Woeginger. We also show that for any n and , there exist sets of points such that G cannot contain a Hamiltonian path if .