On a Conjecture Related to Geometric Routing

We conjecture that any planar 3-connected graph can be embedded in the plane in such a way that for any nodes s and t, there is a path from s to t such that the Euclidean distance to t decreases monotonically along the path. A consequence of this conjecture would be that in any ad hoc network containing such a graph as a subgraph, 2-dimensional virtual coordinates for the nodes can be found for which greedy geographic routing is guaranteed to work. We discuss this conjecture and its equivalent forms. We show a weaker result, namely that for any network containing a 3-connected planar subgraph, 3-dimensional virtual coordinates always exist enabling a form of greedy routing inspired by the simplex method; we provide experimental evidence that this scheme is quite effective in practice. We also propose a rigorous form of face routing based on the Koebe-Andre’ev-Thurston theorem. Finally, we show a result delimiting the applicability of our approach: any 3-connected K3,3-free graph has a planar 3-connected subgraph.