Optimal Polygonal Representation of Planar Graphs
Michael Kaufmann2012, Algorithmica
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Abstract
In this paper, we consider the problem of representing graphs by polygons whose sides touch. We show that at least six sides per polygon are necessary by constructing a class of planar graphs that cannot be represented by pentagons. We also show that the lower bound of six sides is matched by an upper bound of six sides with a linear time algorithm for representing any planar graph by touching hexagons. Moreover, our algorithm produces convex polygons with edges with slopes 0, 1, -1. Fig. 1. Given a drawing of a planar graph(a), we apportion the edges to the endpoints by cutting each edge in half (b), and then apportion the faces to form polygons (c).
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