Drawing Planar Graphs with Few Geometric Primitives

We define the visual complexity of a plane graph drawing to be the number of basic geometric objects needed to represent all its edges. In particular, one object may represent multiple edges (e.g., one needs only one line segment to draw two collinear edges of the same vertex). Let n denote the number of vertices of a graph. We show that trees can be drawn with 3n/4 straight-line segments on a polynomial grid, and with n/2 straight-line segments on a quasi-polynomial grid. Further, we present an algorithm for drawing planar 3-trees with (8n− 17)/3 segments on an O(n)× O(n) grid. This algorithm can also be used with a small modification to draw maximal outerplanar graphs with 3n/2 edges on an O(n)×O(n) grid. We also study the problem of drawing maximal planar graphs with circular arcs and provide an algorithm to draw such graphs using only (5n−11)/3 arcs. This provides a significant improvement over the lower bound of 2n for line segments for a nontrivial graph class.