Euclidean Greedy Drawings of Trees

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 \({\mathbb {R}}^2\) is known for certain graph classes such as 3-connected planar graphs. We completely characterize the trees that admit a greedy embedding in \({\mathbb {R}}^2\). This answers a question by Angelini et al. (Networks 59(3):267–274, 2012) and is a further step in characterizing the graphs that admit Euclidean greedy embeddings.