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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. Such a drawing 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 on a plane, and from every edge to a plane curve on that plane, 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. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa. Plane graphs can be encoded by combinatorial maps. 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. == Kuratowski's and Wagner's theorems == The Polish mathematician Kazimierz Kuratowski provided a characterization of planar graphs in terms of forbidden graphs, now known as Kuratowski's theorem: :A finite graph is planar if and only if it does not contain a subgraph that is a subdivision of ''K''5 (the complete graph on five vertices) or ''K''3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three, also known as the utility graph). A subdivision of a graph results from inserting vertices into edges (for example, changing an edge •——• to •—•—•) zero or more times. Instead of considering subdivisions, Wagner's theorem deals with minors: :A finite graph is planar if and only if it does not have ''K''5 or ''K''3,3 as a minor. Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of "forbidden minors". This is now the Robertson–Seymour theorem, proved in a long series of papers. In the language of this theorem, ''K''5 and ''K''3,3 are the forbidden minors for the class of finite planar graphs. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Planar graph」の詳細全文を読む スポンサード リンク
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