than in unrestricted planar graphs. Upward planar graphs An upward planar graph is a directed acyclic graph that can be drawn in the plane with its edges as non-crossing curves that are consistently oriented in an upward direction. Not every planar directed acyclic graph is upward planar, and it is NP-complete to test whether a given graph is upward planar. Convex planar graphs A planar graph is said to be convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph ). A sufficient condition that a graph can be drawn convexly is that it is a subdivision of a 3-vertex-connected planar graph. Tutte's spring theorem even states that for simple 3-vertex-connected planar graphs the position of the inner vertices can be chosen to be the average of its neighbors. Word-representable planar graphs Word-representable planar graphs include triangle-free planar graphs and, more generally, 3-colourable planar graphs, as well as certain face subdivisions of triangular grid graphs, and certain triangulations of grid-covered cylinder graphs. Theorems Enumeration of planar graphs The asymptotic for the number of (labeled) planar graphs on vertices is , where and . Almost all planar graphs have an exponential number of automorphisms. The number of unlabeled (non-isomorphic) planar graphs on vertices is between and . Other results The four color theorem states that every planar graph is 4-colorable (i.e., 4-partite). Fáry's theorem states that every simple planar graph admits a representation as a planar straight-line graph. A universal point set is a set of points such that every planar graph with n vertices has such an embedding with all vertices in the point set; there exist universal point sets of quadratic size, formed by taking a rectangular subset of the integer lattice. Every simple outerplanar graph admits an embedding in the plane such that all vertices lie on a fixed circle and all edges are straight line segments that lie inside the disk and don't intersect, so n-vertex regular polygons are universal for outerplanar graphs. Scheinerman's conjecture (now a theorem) states that every planar graph can be represented as an intersection graph of line segments in the plane. The planar separator theorem states that every n-vertex planar graph can be partitioned into two subgraphs of size at most 2n/3 by the removal of O() vertices. As a consequence, planar graphs also have treewidth