Steinitz's Theorem (Convex Polytopes)

Steinitz's Theorem, a result of polyhedral combinatorics, characterizes the undirected graphs formed by the vertices and edges of three-dimensional convex polyhedra as exactly the three-vertex-connected planar graphs, so that every convex polyhedron's graph is three-connected and planar, and conversely every three-connected planar graph can be realized as the graph of some convex polyhedron. Branko Grunbaum called it the most important and deepest known result on three-dimensional polytopes, and the theorem appeared in a 1922 publication of Ernst Steinitz, after whom it is named. Because no comparable classification is known in higher dimensions, it gives a purely combinatorial description of three-dimensional polyhedra that has been used to simplify other results, such as Eberhard's theorem on realizing polyhedra with prescribed face types, and to construct three-dimensional visualizations of abstract graphs.