A crossing is a pair of intersecting edges. A graph is embeddable on a surface if its vertices and edges can be arranged on it without any crossing. The genus of a graph is the lowest genus of any surface on which the graph can embed.
A planar graph is one which can be drawn on the (Euclidean) plane without any crossing; and a plane graph, one which is drawn in such fashion. In other words, a planar graph is a graph of genus 0. The example graph is planar; the complete graph on n vertices, for n> 4, is not planar. Also, a tree is necessarily a planar graph.
When a graph is drawn without any crossing, any cycle that surrounds a region without any edges reaching from the cycle into the region forms a face. Two faces on a plane graph are adjacent if they share a common edge. A dual, or planar dual when the context needs to be clarified, G* of a plane graph G is a graph whose vertices represent the faces, including any outerface, of G and are adjacent in G* if and only if their corresponding faces are adjacent in G. The dual of a planar graph is always a planar pseudograph (e.g. consider the dual of a triangle). In the familiar case of a 3-connected simple planar graph G (isomorphic to a convex polyhedron P), the dual G* is also a 3-connected simple planar graph (and isomorphic to the dual polyhedron P*).
Furthermore, since we can establish a sense of "inside" and "outside" on a plane, we can identify an "outermost" region that contains the entire graph if the graph does not cover the entire plane. Such outermost region is called an outer face. An outerplanar graph is one which can be drawn in the planar fashion such that its vertices are all adjacent to the outer face; and an outerplane graph, one which is drawn in such fashion.
The minimum number of crossings that must appear when a graph is drawn on a plane is called the crossing number.
The minimum number of planar graphs needed to cover a graph is the thickness of the graph.
Read more about this topic: Glossary Of Graph Theory
Famous quotes containing the word genus:
“Methinks it would be some advantage to philosophy if men were named merely in the gross, as they are known. It would be necessary only to know the genus and perhaps the race or variety, to know the individual. We are not prepared to believe that every private soldier in a Roman army had a name of his own,—because we have not supposed that he had a character of his own.”
—Henry David Thoreau (1817–1862)