On Planar Greedy Drawings of 3-Connected Planar Graphs

TL;DR

This paper proves that every 3-connected planar graph has a planar greedy drawing, strengthening previous results and advancing towards the conjecture that such graphs can be drawn with convex faces for greedy routing.

Contribution

It establishes that all 3-connected planar graphs admit planar greedy drawings, an important step beyond earlier work and towards convex greedy embeddings.

Findings

Every 3-connected planar graph admits a planar greedy drawing.

This result strengthens previous theorems by Leighton and Moitra.

It serves as an intermediate step towards the convex greedy embedding conjecture.

Abstract

A graph drawing is $\textit{greedy}$ if, for every ordered pair of vertices $(x,y)$, there is a path from $x$ to $y$ such that the Euclidean distance to $y$ decreases monotonically at every vertex of the path. Greedy drawings support a simple geometric routing scheme, in which any node that has to send a packet to a destination "greedily" forwards the packet to any neighbor that is closer to the destination than itself, according to the Euclidean distance in the drawing. In a greedy drawing such a neighbor always exists and hence this routing scheme is guaranteed to succeed. In 2004 Papadimitriou and Ratajczak stated two conjectures related to greedy drawings. The $\textit{greedy embedding conjecture}$ states that every $3$-connected planar graph admits a greedy drawing. The $\textit{convex greedy embedding conjecture}$ asserts that every $3$-connected planar graph admits a planar greedy drawing in which the faces are delimited by convex polygons. In 2008 the greedy embedding conjecture was settled in the positive by Leighton and Moitra. In this paper we prove that every $3$-connected planar graph admits a $\textit{planar}$ greedy drawing. Apart from being a strengthening of Leighton and Moitra's result, this theorem constitutes a natural intermediate step towards a proof of the convex greedy embedding conjecture.