On Convex Greedy Embedding Conjecture for 3-Connected Planar Graphs

A greedy embedding of a graph G = (V, E) into a metric space (X, d) is a function x : V (G) → X such that in the embedding for every pair of non-adjacent vertices x(s), x(t) there exists another vertex x(u) adjacent to x(s) which is closer to x(t) than x(s). This notion of greedy embedding was defined by Papadimitriou and Ratajczak (Theor. Comput. Sci. 2005), where authors 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 proved by Leighton and Moitra (FOCS 2008). However, their algorithm do not result in a drawing that is planar and convex for all 3-connected planar graph in the Euclidean plane. In this work we consider the planar convex greedy embedding conjecture and make some progress. We derive a new characterization of planar convex greedy embedding that given a 3-connected planar graph G = (V, E), an embedding x : V → IR of G is a planar convex greedy embedding if and only if, in the embedding x, weight of the maximum weight spanning tree (T ) and weight of the minimum weight spanning tree (MST) satisfies wt(T )/wt(MST) ≤ (|V | − 1), 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 x : V (G) → X such that for every pair of non-adjacent vertices x(s), x(t) there exists a vertex x(u) adjacent to x(s) such that distance between x(u) and x(t) is at most β times the distance between x(s) and x(t). We show that any three connected planar graph G = (V, E) has a β–weak greedy planar convex embedding in the Euclidean plane with β ∈ [1, 2 √ 2 · d(G)], where d(G) is the ratio of maximum and minimum distance between pair of vertices in the embedding of G. Finally, we also show that this bound is tight for well known Tutte embedding of 3-connected planar graphs in the Euclidean plane which is planar and convex.