We describe a Fast Delivery Guaranteed Face Routing (FDGF) in ad hoc wireless networks. Since it is expensive for wireless nodes to get the whole network topology information, geometric routing decisions should be made locally by nodes using location information of neighboring nodes which are at most k hops away. In this paper, we first define k-local algorithm and obtain two local geometric graphs, i.e., k-Local Delauany Triangulation Graph (k-LDTG) and k-Local Minimum Weight Spanning Tree

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... MWST), which are more efficient than existing definitions. We present problems with existing face routing protocols and propose FDGF to counter possible long delivery delay with high probability. The performance of face routing differs on different planar graphs. We compare face routing characteristics of four different underlying routing graphs which include k-LDTG, Gabriel Graph (GG), Relative Neighbor Graph (RNG) and k-LMWST. Due to different attributes of these graphs, the message delivery delay, routing hop and minimum energy consumption differ greatly. Through experimentation in the NS-2 simulator, we have shown that the proposed face routing protocol achieves 100% delivery ratio on Unit Disk Graph (UDG) when source to destination connection path exists. Face routing on k-LDTG is fast with less relay hops and face routing on k-LMWST is energy efficient. RNG achieves desirable minimum energy consumption attribute, better than all the others when the propagation model is free space and close to k-LMWST when the propagation model is Two Ray Ground.