Planar Graph

Example(s):
- a Maximal Planar Graph,
- a Plane Graph,
- a Butterfly Graph.
- a Complete Graph,
- a Upward Planar Graph,
- …
Counter-Example(s):
- an Apex Graph,
- a Halin Graph,
- a Map Graph,
- an Outerplanar Graph,
- a Toroidal Graph,
- an Undirected Graph.
See: Crossing Edge, Connected Plane Graph, Homeomorphic Relation, Combinatorial Map, Three Utilities Problem, Sprouts Game, Planarization.

References

(Wikipedia, 2011) ⇒ http://en.wikipedia.org/wiki/Planar_graph
- In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. A planar graph already drawn in the plane without edge intersections is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point in 2D space, and from every edge to a plane curve, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Plane graphs can be encoded by combinatorial maps. It is easily seen that a graph that can be drawn on the plane can be drawn on the sphere as well, and vice versa. The equivalence class of topologically equivalent drawings on the sphere is called a planar map. Although a plane graph has an external or unbounded face, none of the faces of a planar map have a particular status. A generalization of planar graphs are graphs which can be drawn on a surface of a given genus. In this terminology, planar graphs have graph genus 0, since the plane (and the sphere) are surfaces of genus 0. See “graph embedding” for other related topics.

(Winter, 2005a) ⇒ Dale Winter. (2005). “Planar Graphs." webpage accessed 2005-Aug-25
- QUOTE: A plane graph is one that has been drawn in the plane in such a way that its edges intersect only at their common end-vertices.