(the second one can be redrawn without intersecting edges by moving one of the diagonal edges to the outside),
while the two graphs shown below are not planar:
It is not possible to redraw these without edge intersections. In fact, these two are the smallest non-planar graphs, a consequence of the characterization below.
- A finite graph is planar if and only if it does not contain a subgraph that is an expansion of K5 (the complete graph on 5 vertices) or K3,3 (complete bipartite graph on six vertices, three of which connect to each of the other three).
- A finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to K5 or K3,3.
- A finite graph is planar if and only if it does not have K5 or K3,3 as a minor.
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 n vertices, it is possible to determine in time O(n) whether the graph is planar or not. Furthermore, for two planar graphs with n vertices, it is possible to determine in time O(n) whether they are isomorphic or not.
- explain these algorithms
Euler'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), then
- v − e + f = 2,
In a finite connected simple planar graph, any face (except possibly the outer one) is bounded by at least three edges and every edge touches at most two faces; using Euler's formula, one can then show that these graphs are sparse in the sense that e ≤ 3v - 6 if v ≥ 3.
A simple graph is called maximal planar if it is planar but adding any edge would destroy that properties. All faces (even the outer one) are then bounded by three edges, explaining the alternative term triangular for these graphs. If a triangular graph has v vertices with v > 2, then it has precisely 3v-6 edges and 2v-4 faces.
Note that Euler's formula is also valid for simple polyhedra. This is no coincidence: every simple polyhedron can be turned into a connected simple planar graph by using the polyhedron's vertices as vertices of the graph and the polyhedron's edges as edges of the graph. The faces of the resulting planar graph then correspond to the faces of the polyhedron. For example, the second planar graph shown above corresponds to a tetrahedron. Not every connected simple planar graph belongs to a simple polyhedron in this fashion: the trees do not, for example. A theorem of Steinitz says that the planar graphs formed from convex polyhedra (equivalently: those formed from simple polyhedra) are precisely the finite 3-connected simple planar graphs.
A graph is called outerplanar if it has an embedding in the plane such that the vertices lie on a fixed circle and the edges lie inside the disk of the circle and don't intersect. Equivalently, there is some face that includes every vertex. Obviously, every outerplanar graph is planar, but the converse is not true: the second example graph shown above (K4) is planar but not outerplanar. This is the smallest non-outerplanar graph: a theorem similar to Kuratowski's states that a finite graph is outerplanar if and only if it does not contain a subgraph that is an expansion of K4 (the full graph on 4 vertices) or of K2,3 (five vertices, 2 of which connnected to each of the other three for a total of 6 edges).
It can be shown that every simple planar graph admits an embedding in the plane such that all edges are straight line segments which don't intersect. Similarly, 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.