We propose several novel localized algorithms to construct energy efficient routing structures for homogeneous wireless ad hoc networks, where all nodes have same maximum transmission ranges. Our first structure has the following attractive properties: (1) It is energy efficient: given any two nodes u and v, there is a path connecting them in the structure with total energy cost at most = 1 1-(2 sin k ) times of the energy cost of any path connecting them in original communication graph; (2) Its node degree is bounded from above by a positive constant k + 5 where k > 6 is an adjustable parameter; (3) It is a planar structure, which enables several localized routing algorithms; (4) It can be constructed and maintained locally and dynamically. Moreover, by assuming that the node ID and its position can be represented in O(log n) bits each for a wireless network of n nodes, we show that the structure can be constructed using at most 24n messages, where each message is O(log n) bits. O...