# Skeleton

The k -skeleton of a polytope ${\displaystyle {\mathcal {P}}}$ is the set of all elements of ${\displaystyle {\mathcal {P}}}$ 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

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 ${\displaystyle K_{6}}$ 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.[2][3] 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).