# Skeleton

The k -**skeleton** of a polytope is the set of all elements of with rank at most equal to k .^{[1]}

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.^{[1]} 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[edit | edit source]

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 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.^{[2]}^{[3]} Such graphs are called *polyhedral graphs*.

## References[edit | edit source]

- ↑
^{1.0}^{1.1}Kalai (2017:505) - ↑ Kalai (2017:506)
- ↑ Steinitz (1922:22)

## Bibligraphy[edit | edit source]

- 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).