One of the new challenges facing research in wireless networks is the design of algorithms and protocols that are energy aware. A good example is the minimum-energy broadcast routing problem for a static network in the plane, which attracted a great deal of attention these past years. The problem is NP-hard and its approximation ratio complexity is a solution proved to be within a factor 6 of the optimal, based on finding a Minimum Spanning Tree of the static planar network. In this paper, we use for the first time the evolving graph combinatorial model as a tool to prove an NP-Completeness result, namely that computing a Minimum Spanning Tree of a planar network in the presence of mobility is actually NP-Complete. This result implies that the above approximation solution cannot be used in dynamic wireless networks. On the positive side, we give a polynomial-time algorithm to build a rooted spanning tree of an on/off network, that minimizes the maximum energy used by any one node. Such tree could then be used in order to maximize the life-time of wireless communication networks.
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