Skeleton

Definition
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 it is the most commonly used 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

 * A graph is the $1$-skeleton of a 3-polytope iff it is planar.