Succinct Strictly Convex Greedy Drawing of 3-Connected Plane Graphs
Geometric routing by using virtual locations is an elegant way for solving network routing problems. Greedy routing, where a message is simply forwarded to a neighbor that is closer to the destination, is a simple form of geometric routing. Papadimitriou and Ratajczak conjectured that every 3-connected plane graph has a greedy drawing in the \(\mathcal R^2\) plane . Leighton and Moitra settled this conjecture positively in . However, their drawings have two major drawbacks: (1) their drawings are not necessarily planar; and (2) Ω(nlogn) bits are needed to represent the coordinates of their drawings, which is too large for routing algorithms for wireless networks. Recently, He and Zhang  showed that every triangulated plane graph has a succinct (using O(logn) bit coordinates) greedy drawing in \(\mathcal R^2\) plane with respect to a metric function derived from Schnyder realizer. However, their method fails for 3-connected plane graphs. In this paper, we show that every 3-connected plane graph has drawing in the \(\mathcal R^2\) plane, that is succinct, planar, strictly convex, and is greedy with respect to a metric function based on parameters derived from Schnyder wood.
KeywordsPlane Graph Euclidean Plane Consecutive Vertex Schnyder Wood Edge Pattern
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