On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs

Abstract

In the context of geographic routing, Papadimitriou and Ratajczak conjectured that every 3-connected planar graph has a greedy embedding (possibly planar and convex) in the Euclidean plane. Recently, greedy embedding conjecture has been resolved, though the construction do not result in a drawing that is planar and convex. In this work we consider the planar convex greedy embedding conjecture and make some progress. We show that in planar convex greedy embedding of a graph, weight of the maximum weight spanning tree (T) and weight of the minimum weight spanning tree (MST) satisfies ${\sf wt}(T)/{\sf wt}(\sf {MST}) \leq \left(|{\it V}|-1\right)^{1 - \delta}$, for some 0 < δ ≤ 1. In order to present this result we define a notion of weak greedy embedding. For β ≥ 1 a β–weak greedy embedding of a graph is a planar embedding such that local optima is bounded by β. We also show that any three connected planar graph has a β–weak greedy planar convex embedding in the Euclidean plane with $\beta \in [1,2\sqrt{2} \cdot d(G)]$, where d(G) is the ratio of maximum and minimum distance between pair of vertices in the embedding of G, and this bound is tight.