• DOI:10.2307/3615008
• Corpus ID: 6663488

Convex Polytopes

@inproceedings{Grunbaum1967ConvexP,
  title={Convex Polytopes},
  author={Branko Grunbaum and Geoffrey C. Shephard},
  year={1967}
}

• B. Grunbaum, G. C. Shephard
• Published 1967
• Mathematics

Graphs, Graphs and Realizations Before proceeding to the graph version of Euler’s formula, some notation will be introduced. A (finite) abstract graph G consists of two sets, the set of vertices V = V(G) = {1, . . . , v} and the set of edges E = E(G) which consists of two-element subsets of V. If {u, w} is an edge, we write also uw for it and call w a neighbour of u and vice versa. If a vertex is contained in an edge or an edge contains a vertex then the vertex and the edge are incident. If two edges have a vertex in common, they are incident at this vertex. A path is a sequence of edges such that each edge is incident with the preceding edge at one vertex and with the next edge at the other vertex. We may write a path also as a sequence w1w2 · · · wk of vertices. We say that it connects the first and the last vertex of this sequence. A path is a cycle if the first and the last vertex coincide. A cycle which consists of k edges is a k-cycle. G is connected if any two distinct of its vertices are connected by a path. If G has at least k + 1 vertices, it is kconnected if any two distinct vertices can be connected by k paths which, pairwise, have only these vertices in common. With some effort it can be shown that this is equivalent to the following. The graph which is obtained from G by deleting any set of k − 1 vertices and the edges incident with these, is still connected. More geometrically, by definition, a (finite) graph consists of a finite point set in some space, the set of vertices and, for each of a set of pairs of distinct vertices, of a continuous curve which connects these vertices and contains no vertex in its relative interior. These curves form the set of edges. The notions and notations which were defined earlier for abstract graphs are defined for graphs in the obvious way. Two graphs G,H, one or both of which may be abstract, are isomorphic if there is a 262 Convex Polytopes bijection between V(G) and V(H) which maps (endpoints of) edges onto (endpoints of) edges in both directions. A graph in E2 (or C or C ∪ {∞}) is planar if no two edges of it cross. More generally, an abstract graph or a graph is called planar if it is isomorphic to a planar graph in E2. The isomorphic image in E2 then is a planar realization of the given, possibly abstract graph. There are simple criteria for planarity. Let G be a planar connected graph in E2. Omitting from E2 the vertices and edges of G leaves a finite system of connected open sets, the countries of G, often called domains of G. Since G is connected, the bounded countries are simply connected. If G is considered as embedded into C ∪ {∞}, also the unbounded country is simply connected. It is easy to see that each planar graph in E2 is isomorphic to a planar graph in E2 the edges of which are polygonal curves. The edge graph G(P) of a proper convex polytope P is the graph consisting of the vertices and the edges of P . If dim P = 3, then G(P) is planar. For more information on geometric graph theory, see Nishizeki and Chiba [772], Bollobás [142], Mohar and Thomassen [748] and Felsner [332]. Euler’s Formula for Graphs in E2 We will show the following result. Theorem 15.2. Let G be a connected planar graph in E2 with v vertices, e edges and f countries. Then v − e + f = 2. Proof. Define the Euler characteristic χ(H) of a connected planar graph H in E2 as the number of vertices minus the number of edges plus the number of countries of H, including the unbounded country. For simplicity, we assume that all graphs considered have polygonal curves as edges. The following proposition is well-known. (2) Let a graph K in E2 arise from a connected planar graph H in E2 by one of the following two operations: (i) Consider a polygonal curve connecting a vertex of H with a point in a country of H and which contains no vertex or point on an edge of H in its relative interior. Add this curve to the edges of H and its endpoint to the vertices. (ii) Consider a polygonal curve connecting two vertices of H and which contains no vertex or point on an edge of H in its relative interior. Add this curve to the edges of H. Then χ(K) = χ(H). Since G is connected, it can be constructed by finitely many operations as described in (2), beginning with a graph consisting of a single vertex which has Euler characteristic 2. Proposition (2) then shows that χ(G) = 2. '( 15 Combinatorial Theory of Convex Polytopes 263 Corollaries of Euler’s Formula for Graphs in E2 The following consequences of Euler’s formula will be needed later. Corollary 15.1. Let H be a planar graph in E2 with v vertices, e edges and f countries. If H contains no 3-cycle, then