Graph of a polytope

Summary

In polytope theory, the edge graph (also known as vertex-edge graph or just graph) of a polytope is a combinatorial graph whose vertices and edges correspond directly to the vertices and edges of the polytope. As a purely combinatorial object, the edge graph encodes incidence information, capturing which vertices are connected by edges, but it does not retain geometric data such as vertex positions or edge lengths. Further common names for the edge graph are skeleton and 1-skeleton, though some authors reserve these terms for the geometric embedding formed by the vertices and edges in the polytope's ambient space. There is no universally agreed upon notation for the edge graph of a polytope $P$. Common notations include $G_P$, $G(P)$ or $\operatorname{skel}(P)$.