Every Schnyder Drawing is a Greedy Embedding

TL;DR

This paper proves that all Schnyder drawings of planar graphs serve as greedy embeddings, ensuring successful geographic routing by leveraging a generalized greedy routing framework.

Contribution

It establishes that every Schnyder drawing is a greedy embedding, connecting classical graph drawing methods with geographic routing guarantees.

Findings

All Schnyder drawings are greedy embeddings.

Greedy routing always guarantees delivery on Schnyder drawings.

Generalized greedy routing framework applied to Schnyder drawings.

Abstract

Geographic routing is a routing paradigm, which uses geographic coordinates of network nodes to determine routes. Greedy routing, the simplest form of geographic routing forwards a packet to the closest neighbor towards the destination. A greedy embedding is a embedding of a graph on a geometric space such that greedy routing always guarantees delivery. A Schnyder drawing is a classical way to draw a planar graph. In this manuscript, we show that every Schnyder drawing is a greedy embedding, based on a generalized definition of greedy routing.