Drawing Graphs with Vertices and Edges in Convex PositionGarcíaMarco, Ignacio and Knauer, Kolja (2015) Drawing Graphs with Vertices and Edges in Convex Position. In: Graph Drawing and Network Visualization: 23rd International Symposium, GD 2015, September 2426, 2015 , pp. 348359(Official URL: http://dx.doi.org/10.1007/9783319272610_29). Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/9783319272610_29
AbstractA graph has strong convex dimension 2, if it admits a straightline drawing in the plane such that its vertices are in convex position and the midpoints of its edges are also in convex position. Halman, Onn, and Rothblum conjectured that graphs of strong convex dimension 2 are planar and therefore have at most 3n−6 edges. We prove that all such graphs have at most 2n−3 edges while on the other hand we present a class of nonplanar graphs of strong convex dimension 2. We also give lower bounds on the maximum number of edges a graph of strong convex dimension 2 can have and discuss variants of this graph class. We apply our results to questions about large convexly independent sets in Minkowski sums of planar point sets, that have been of interest in recent years.
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