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.
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Gansner, E.R., Hu, Y.F., Kaufmann, M., Kobourov, S.G. (2010). Optimal Polygonal Representation of Planar Graphs. In: López-Ortiz, A. (eds) LATIN 2010: Theoretical Informatics. LATIN 2010. Lecture Notes in Computer Science, vol 6034. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12200-2_37
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DOI: https://doi.org/10.1007/978-3-642-12200-2_37
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