ABSTRACT
Topological graph theory was first discovered in 1736 by L. Euler and then was dormant for 191 years. The subject was revived when Kasimir Kuratowski found a criterion for a graph to be planar. All the known criteria for planarity are presented. These include the theorems of Kuratowski and K. Wagner, which characterize planar graphs in terms of forbidden subgraphs, H. Whitney's result in terms of the existence of a combinatorial dual, and S. MacLane's description of the existence of a prescribed cycle basis. A graph is planar if it can be embedded in the plane; a plane graph has already been embedded in the plane. The subject of planar graphs was discovered by Euler in his investigation of polyhedra. A maximal planar graph is one to which no line can be added without losing planarity. Clearly, a graph is planar if and only if each of its components is planar.
