Euclidean Greedy Drawings of Trees

TL;DR

This paper characterizes exactly which trees can be greedily embedded in the Euclidean plane, advancing understanding of geometric routing strategies in wireless networks.

Contribution

It provides a complete characterization of trees that admit Euclidean greedy embeddings, answering an open question from prior research.

Findings

Identifies the class of trees with greedy embeddings in R^2

Answers a previously open question in graph drawing

Advances the theoretical understanding of geometric routing

Abstract

Greedy embedding (or drawing) is a simple and efficient strategy to route messages in wireless sensor networks. For each source-destination pair of nodes s, t in a greedy embedding there is always a neighbor u of s that is closer to t according to some distance metric. The existence of greedy embeddings in the Euclidean plane R^2 is known for certain graph classes such as 3-connected planar graphs. We completely characterize the trees that admit a greedy embedding in R^2. This answers a question by Angelini et al. (Graph Drawing 2009) and is a further step in characterizing the graphs that admit Euclidean greedy embeddings.