Broadcasting, in the context of ad-hoc networks, is a costly operation, and thus topology control has been proposed to achieve efficient broadcasting with low interference and low energy consumption. By topology control, each node optimizes its transmission power by maintaining network connectivity in a localized manner. Local Minimum Spanning Tree (LMST) is the state-of-the-art topology control algorithm, which has been proven to provide satisfactory performance. However, LMST almost always results in a 1-connected network, without redundancy to tolerate external factors. In this paper, we propose Local Tree-based Reliable Topology (LTRT), which is mathematically proven to guarantee k-edge connectivity while preserving the features of LMST. LTRT can be easily constructed with a low computational complexity of O(k(m + n log n)), where k is the connectivity of the resulting topology, n is the number of neighboring nodes, and m is the number of edges. Simulation results have demonstrated the efficiency of LTRT and its superiority over other localized algorithms.