Discrete & Computational Geometry

, Volume 44, Issue 3, pp 686–705

# Some Results on Greedy Embeddings in Metric Spaces

• Tom Leighton
• Ankur Moitra
Article

## Abstract

Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abstraction of this model in which nodes are assigned virtual coordinates in a metric space, and these coordinates are used to perform point-to-point routing.

Here we resolve a conjecture of Papadimitriou and Ratajczak that every 3-connected planar graph admits a greedy embedding into the Euclidean plane. This immediately implies that all 3-connected graphs that exclude K 3,3 as a minor admit a greedy embedding into the Euclidean plane. We also prove a combinatorial condition that guarantees nonembeddability. We use this result to construct graphs that can be greedily embedded into the Euclidean plane, but for which no spanning tree admits such an embedding.

## Keywords

Greedy embedding Papadimitriou–Ratajczak conjecture Christmas cactus graph Excluded minor

## References

1. 1.
Barnette, D.: Trees in polyhedral graphs. Can. J. Math. 18, 731–736 (1966)
2. 2.
Bose, P., Morin, P., Stojmenovic, I., Urrutia, J.: Routing with guaranteed delivery in ad hoc wireless networks. Wirel. Netw. 7, 609–616 (2001)
3. 3.
Dhandapani, R.: Greedy drawings of triangulations. In: Symposium on Discrete Algorithms, pp. 102–111 (2008) Google Scholar
4. 4.
Eppstein, D., Goodrich, M.T.: Succinct greedy graph drawing in the hyperbolic plane. In: Graph Drawing, pp. 14–25 (2008) Google Scholar
5. 5.
Gao, Z., Richter, R.: 2-walks in circuit graphs. J. Comb. Theory Ser. B 62, 259–267 (1994)
6. 6.
Goodrich, M.T., Strash, D.: Succinct greedy geometric routings in 2. In: CoRR (2008) Google Scholar
7. 7.
Karp, B., Kung, H.T.: GPSR: greedy perimeter stateless routing for wireless networks. In: Proceedings of the 6th Annual International Conference on Mobile Computing and Networking, pp. 243–254 (2000) Google Scholar
8. 8.
Kleinberg, R.: Geographic routing in hyperbolic space. In: INFOCOM 2007, 26th IEEE Conference on Computer Communications, pp. 1902–1909 (2007) Google Scholar
9. 9.
Papadimitriou, C., Ratajczak, D.: On a conjecture related to geometric routing. Theor. Comput. Sci. 244, 3–14 (2005)
10. 10.
Rao, A., Papadimitriou, C., Shenker, S., Stoica, I.: Geographic routing without location information. In: Proceedings of the 9th Annual International Conference on Mobile Computing and Networking, pp. 96–108 (2003) Google Scholar