A planar graph is a graph which can be drawn on a plane (a flat 2-d surface) or on a sphere, with no edges crossing. When drawn on a sphere, the edges divide its area in a number of regions called faces (or “countries”, in the context of map coloring). When drawn on a plane, there is one outer country taking up all the space outside the drawing. Every graph drawn on a sphere can be drawn on a plane (puncture the sphere in the interior of any one of the countries) and vice versa. Statements on map coloring are often simpler in terms of a spherical map because the outer country is no longer a special case.
The number of faces (countries) equals where is the cyclomatic number, (where is the number of edges, the number of vertices, and the number of connected components of the graph). All this holds equally for planar multigraphs and pseudographs.
No complete graphs above are planar. , drawn without crossings, looks like :
Hence it is planar (try this for .)
A drawing of a planar graph is a drawing in which each edge is drawn as a line segment. Every planar graph has a drawing. This result was found independently by Wagner, Fáry and Stein. Schnyder improved this further by showing how to draw any planar graph with vertices on an integer grid of area.
Let be a topological manifold. Then a graph on is a pair , where
is a multigraph,
is a function from the graph topology of into , and
is a homeomorphism onto its image.
A plane graph is a graph on .
A planar graph is a graph that has an embedding making into a plane graph.
The most usual question for which this definition finds a use is “can the following graph be made into a graph on ?”. When is the plane, this is usually phrased as “Is a planar graph?”. Wagner’s theorem provides a criterion for answering this question. When is a torus, the answer changes: the complete bipartite graph can be made into a graph on the torus.
A graph on a manifold has a notion of “face” as well as the usual graph notions of vertex and edge.
|Date of creation||2013-03-22 12:17:37|
|Last modified on||2013-03-22 12:17:37|
|Last modified by||archibal (4430)|