Skeleton

The k -skeleton of a polytope ${\mathcal {P}}$ is the set of all elements of ${\mathcal {P}}$ with rank at most equal to k .

The 1 -skeleton of a polytope can be thought of as a graph with its vertex set being the vertices of the polytope and its edge set being the edges of the polytope. Thus two vertices are adjacent iff there is an edge in the polytope which is incident on both of them. This graph may be referenced by a number of names. Since the 1 -skeleton is the most commonly used k -skeleton it may be referred to as "the skeleton" of a polytope without a qualifying numeral. It may also be called simply the graph of a polytope or the vertex adjacency graph of a polytope.

Properties

All vertices of the 1-skeleton of a Euclidean polytope of rank d  have a degree of at least d . This is not necessarily true of abstract polytopes, maniplexes or polytopes in other spaces.

Polytopes of different ranks can have the same 1-skeleton. The complete graph $K_{6}$ is the 1-skeleton of both a 4-polytope and 5-polytope.

A finite graph is the 1 -skeleton of a finite convex polyhedron iff the graph is 3-vertex-connected and planar. Such graphs are called polyhedral graphs.

Bibligraphy

• Kalai, Gil (10 August 2017). "Polytope Skeletons and Paths" (PDF).
• Steinitz, Ernst (1922). "Polyeder und Raumeinteilungen" [Polyhedra and Divisions of Space]. Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometries) [Encyclopedia of the Mathematical Sciences, Volume 3 (Geometry)] (in German).