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In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of three-dimensional convex polyhedra: they are exactly the (simple) 3-vertex-connected planar graphs (with at least four vertices).〔''Lectures on Polytopes'', by Günter M. Ziegler (1995) ISBN 0-387-94365-X , Chapter 4 "Steinitz' Theorem for 3-Polytopes", p.103.〕〔Branko Grünbaum, ''Convex Polytopes'', 2nd edition, prepared by Volker Kaibel, Victor Klee, and Günter M. Ziegler, 2003, ISBN 0-387-40409-0, ISBN 978-0-387-40409-7, 466pp.〕 That is, every convex polyhedron forms a 3-connected planar graph, and every 3-connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3-connected planar graphs are also known as polyhedral graphs. Steinitz's theorem is named after Ernst Steinitz, who submitted its first proof for publication in 1916.〔.〕 Branko Grünbaum has called this theorem “the most important and deepest known result on 3-polytopes.”〔 The name "Steinitz's theorem" has also been applied to other results of Steinitz: * the Steinitz exchange lemma implying that each basis of a vector space has the same number of vectors,〔.〕 * the theorem that if the convex hull of a point set contains a unit sphere, then the convex hull of a finite subset of the point contains a smaller concentric sphere,〔.〕 and * Steinitz's vectorial generalization of the Riemann series theorem on the rearrangements of conditionally convergent series.〔.〕〔 〕〔 〕〔 〕 ==Definitions and statement of the theorem== An undirected graph is a system of vertices and edges, each edge connecting two of the vertices. From any polyhedron one can form a graph, by letting the vertices of the graph correspond to the vertices of the polyhedron and by connecting any two graph vertices by an edge whenever the corresponding two polyhedron vertices are the endpoints of an edge of the polyhedron. This graph is known as the skeleton of the polyhedron. A graph is planar if it can be drawn with its vertices as points in the Euclidean plane, and its edges as curves that connect these points, such that no two edge curves cross each other and such that the point representing a vertex lies on the curve representing an edge only when the vertex is an endpoint of the edge. By Fáry's theorem, it is sufficient to consider only planar drawings in which the curves representing the edges are line segments. A graph is 3-connected if, after the removal of any two of its vertices, any other pair of vertices remain connected by a path. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Steinitz's theorem」の詳細全文を読む スポンサード リンク
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