Canonical Orders and Schnyder Realizers

Every planar graph has a crossings-free drawing in the plane. Formally, a straight-line drawing of a planar graph G is one where each vertex is placed at a point in the plane and each edge is represented by a straight-line segment between the two corresponding points such that no two edges cross each other, except possibly at their common endpoints. A straight-line grid drawing of G is a straight-line drawing of G where each vertex of G is placed on an integer grid point. The area for such a drawing is defined by the minimum-area axis-aligned rectangle, or bounding box, that contains the drawing. proved independently that every planar graph has a straight-line drawing. It was not until 1990 that the first algorithms for drawing a planar graph on a grid of polynomial area were developed. The concepts of canonical orders [4] and Schnyder realizers [9] were independently introduced for the purpose of efficiently computing straight-line grid drawings on the O(n) × O(n) grid. These two seemingly very different combinatorial structures turn out to be closely related and have since been used in many different problems and in many applications.