Optimal polygonal representation of planar graphs

Authors:
E. R. Gansner;Y. F. Hu;M. Kaufmann;S. G. Kobourov
Affiliations:
ATST Research Labs, Florham Park, NJ;ATST Research Labs, Florham Park, NJ;Wilhelm-Schickhard-Institut for Computer Science, Tübingen University;Dept. of Computer Science, University of Arizona
Venue:
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Year:
2010

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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.