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In this paper, the authors provide a thorough theoretical study on delivery guarantees, loop-free operation, and worst-case behavior of face and combined greedy-face routing. They show that under specific planar topology control schemes, recovery from a greedy routing failure is always possible without changing between any adjacent faces. Guaranteed delivery then follows from guaranteed recovery while traversing the very first face. In arbitrary planar graphs, however, a proper face selection mechanism is of importance since recovery from a greedy routing failure may require visiting a sequence of faces before greedy routing can be restarted again. They provide complete and formal proofs that several proposed face routing and combined greedy-face routing schemes guarantee message delivery in specific planar graph classes or even in arbitrary planar graphs.
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