Discrete & Computational Geometry

, Volume 44, Issue 3, pp 686–705

Some Results on Greedy Embeddings in Metric Spaces

Article

DOI: 10.1007/s00454-009-9227-6

Cite this article as:
Leighton, T. & Moitra, A. Discrete Comput Geom (2010) 44: 686. doi:10.1007/s00454-009-9227-6

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 K3,3 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 

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Math. Department and CSAILMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.EECS Department and CSAILMassachusetts Institute of TechnologyCambridgeUSA

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