We propose a family of structures, namely, k-localized minimum spanning tree (LMSTk) for topology control and broadcasting in wireless ad hoc networks. We give an efficient localized method to construct LMSTk using only O(n) messages under the local-broadcast communication model, i.e., the signal sent by each node would be received by all nodes within the node's transmission range. We also analytically prove that the node degree of the structure LMSTk is at most 6, LMSTk is connected and planar and, more importantly, the total edge length of the LMSTk is within a constant factor of that of the minimum spanning tree when k≥2 (called low weighted hereafter). We then propose another low weighted structure, called Incident MST and RNG Graph (IMRG), that can be locally constructed using at most 13n messages under the local broadcast communication model. Test results are corroborated in the simulation study. We study the performance of our structures in terms of the total power consumption for broadcasting, the maximum node power needed to maintain the network connectivity. We theoretically prove that our structures are asymptotically the best possible for broadcasting among all locally constructed structures. Our simulations show that our new structures outperform previous locally constructed structures in terms of broadcasting and power assignment for connectivity.