Some Results 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 $x : V(G) \to X$ 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 $x: V \to \bbbr^2$ 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 ($\func{MST}$) satisfies $\WT(T)/\WT(\func{MST}) \leq (\card{V}-1)^{1 - \delta}$, for some $0 < \delta \leq 1$.

💡 Research Summary

The paper addresses the long‑standing planar 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 mapping (x:V\rightarrow X) such that for every pair of non‑adjacent vertices (s,t) there exists a neighbor (u) of (s) with (d(x(u),x(t))<d(x(s),x(t))). Papadimitriou and Ratajczak conjectured that every 3‑connected planar graph admits a greedy embedding in the Euclidean plane, and Leighton and Moitra proved the existence of a greedy embedding for all such graphs in 2008. However, the Leighton‑Moitra construction does not guarantee that the embedding is planar (i.e., edges do not cross) nor that all faces are convex polygons.

The authors focus on the stronger “planar convex greedy embedding” (PCGE) – an embedding that is simultaneously planar, has all interior faces convex, and satisfies the greedy property. They introduce a novel quantitative condition based on the weights of two spanning trees in the embedded graph: the maximum‑weight spanning tree (T) (where edge weight equals Euclidean length) and the minimum‑spanning tree (\text{MST}). Their main theorem states that a planar embedding (x) of a 3‑connected planar graph (G) is a PCGE if and only if there exists a constant (\delta) with (0<\delta\le 1) such that

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