Planar graph

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 planar graph is one that can be drawn on a plane in such a way that there are no "edge crossings," i.e. edges intersect only at their common vertices.

(Robertson & Seymour, 1986) ⇒ Neil Robertson, and P D Seymour. (1986). “Graph minors. V. Excluding a planar graph.” In: Journal of Combinatorial Theory, Series B, 41(1). doi:10.1016/0095-8956(86)90030-4
- CITED BY: http://scholar.google.com/scholar?&q=%22Graph minors. V. Excluding a planar graph%22+1986

(Scheinerman & West, 1983) ⇒ Edward R. Scheinerman, and Douglas B. West. (1983). “The interval number of a planar graph: Three intervals suffice.” In: Journal of Combinatorial Theory, Series B, 35(3) [doi>10.1016/0095-8956(83)90050-3]
- QUOTE: … A planar graph is outerplanar if it can be embedded in the plane so that every vertex is on the unbounded face; for outerplanar graphs we discuss only embeddings of that type, in which case G = G. Given a plane graph G, its dual graph G* has a vertex for each face of G and an ...

(Auslander & Parter, 1961) ⇒ L. Auslander, S. Parter. (1961). “On Imbedding Graphs in the Sphere.” In: Indiana Univ. Math. J. 10 No. 3.

(Tutte, 1956) ⇒ W. T. Tutte. (1956). “A Theorem on Planar Graphs.” In: Transactions of the American Mathematical Society, 82(1).
- … A planar graph is a finite set of simple closed arcs, called edges, in the 2-sphere such that any point of intersection of two distinct members of the set is an end of both of them. The vertices of a planar graph are the ends of its edges. ...