Optimal Polygonal Representation of Planar Graphs

TL;DR

This paper establishes that representing planar graphs with polygons requires at least six sides per polygon, and provides a linear-time algorithm to achieve such representations with convex hexagons, optimizing for simplicity and grid placement.

Contribution

The paper proves a lower bound of six sides per polygon for planar graph representation and introduces a linear-time algorithm for convex hexagon representations.

Findings

At least six sides are necessary for polygonal representation of certain planar graphs.

A linear-time algorithm exists for representing any planar graph with convex hexagons.

The algorithm produces polygons with edges of at most three slopes on an O(n) grid.

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 having at most three slopes and with all vertices lying on an O(n)xO(n) grid.