of "forbidden minors". This is now the Robertson–Seymour theorem, proved in a long series of papers. In the language of this theorem, and are the forbidden minors for the class of finite planar graphs. Other criteria In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. However, there exist fast algorithms for this problem: for a graph with vertices, it is possible to determine in time (linear time) whether the graph may be planar or not (see planarity testing). For a simple, connected, planar graph with vertices and edges and faces, the following simple conditions hold for : Theorem 1. ; Theorem 2. If there are no cycles of length 3, then . Theorem 3. . In this sense, planar graphs are sparse graphs, in that they have only edges, asymptotically smaller than the maximum . The graph , for example, has 6 vertices, 9 edges, and no cycles of length 3. Therefore, by Theorem 2, it cannot be planar. These theorems provide necessary conditions for planarity that are not sufficient conditions, and therefore can only be used to prove a graph is not planar, not that it is planar. If both theorem 1 and 2 fail, other methods may be used. Whitney's planarity criterion gives a characterization based on the existence of an algebraic dual; Mac Lane's planarity criterion gives an algebraic characterization of finite planar graphs, via their cycle spaces; The Fraysseix–Rosenstiehl planarity criterion gives a characterization based on the existence of a bipartition of the cotree edges of a depth-first search tree. It is central to the left-right planarity testing algorithm; Schnyder's theorem gives a characterization of planarity in terms of partial order dimension; Colin de Verdière's planarity criterion gives a characterization based on the maximum multiplicity of the second eigenvalue of certain Schrödinger operators defined by the graph. The Hanani–Tutte theorem states that a graph is planar if and only if it has a drawing in which each independent pair of edges crosses an even number of times; it can be used to characterize the planar graphs via a system of equations modulo 2. Properties Euler's formula Euler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and is the number of vertices, is the number of edges and is the number of faces (regions bounded by edges, including