Planar disk graph proof6/7/2023 ![]() In general, if the property holds for all planar graphs of f faces, any change to the graph that creates an additional face while keeping the graph planar would keep v – e + f an invariant. Main article: Euler characteristic § Plane graphsĮuler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), thenĪs an illustration, in the butterfly graph given above, v = 5, e = 6 and f = 3. 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.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.Schnyder's theorem gives a characterization of planarity in terms of partial order dimension.It is central to the left-right planarity testing algorithm 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.Mac Lane's planarity criterion gives an algebraic characterization of finite planar graphs, via their cycle spaces.Whitney's planarity criterion gives a characterization based on the existence of an algebraic dual.If both theorem 1 and 2 fail, other methods may be used. 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. Therefore, by Theorem 2, it cannot be planar. The graph K 3,3, for example, has 6 vertices, 9 edges, and no cycles of length 3. In this sense, planar graphs are sparse graphs, in that they have only O( v) edges, asymptotically smaller than the maximum O( v 2). If there are no cycles of length 3, then e ≤ 2 v – 4. However, there exist fast algorithms for this problem: for a graph with n vertices, it is possible to determine in time O( n) (linear time) whether the graph may be planar or not (see planarity testing).įor a simple, connected, planar graph with v vertices and e edges and f faces, the following simple conditions hold for v ≥ 3: In practice, it is difficult to use Kuratowski's criterion to quickly decide whether a given graph is planar. In the language of this theorem, K 5 and K 3,3 are the forbidden minors for the class of finite planar graphs. This is now the Robertson–Seymour theorem, proved in a long series of papers. Klaus Wagner asked more generally whether any minor-closed class of graphs is determined by a finite set of " forbidden minors". ![]() Planarity criteria Kuratowski's and Wagner's theoremsĪn animation showing that the Petersen graph contains a minor isomorphic to the K 3,3 graph, and is therefore non-planar See " graph embedding" for other related topics. In this terminology, planar graphs have genus 0, since the plane (and the sphere) are surfaces of genus 0. Planar graphs generalize to graphs drawable on a surface of a given genus. Although a plane graph has an external or unbounded face, none of the faces of a planar map has a particular status. Plane graphs can be encoded by combinatorial maps or rotation systems.Īn equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a planar map. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.Įvery graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Such a drawing is called a plane graph or planar embedding of the graph. In other words, it can be drawn in such a way that no edges cross each other. In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. ![]() Graph that can be embedded in the plane Example graphs
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