Transitions in geometric minimum spanning trees
Discrete & Computational Geometry - Special issue on ACM symposium on computational geometry, North Conway
Geometric ad-hoc routing: of theory and practice
Proceedings of the twenty-second annual symposium on Principles of distributed computing
Geographic routing without location information
Proceedings of the 9th annual international conference on Mobile computing and networking
On a conjecture related to geometric routing
Theoretical Computer Science - Algorithmic aspects of wireless sensor networks
Greedy drawings of triangulations
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On convex embeddings of planar 3-connected graphs
Journal of Graph Theory
Journal of Graph Theory
Some Results on Greedy Embeddings in Metric Spaces
FOCS '08 Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science
Graph Theory
On convex greedy embedding conjecture for 3-connected planar graphs
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
On convex greedy embedding conjecture for 3-connected planar graphs
FCT'09 Proceedings of the 17th international conference on Fundamentals of computation theory
On succinct convex greedy drawing of 3-connected plane graphs
Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms
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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 wt(T)/wt(MST) ≤ (|V| - 1)1-δ, for some 0 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.