The skeleton of a convex polytope

Takayuki Hibi, Aki Mori

Published 2023-07-17Version 1

Let ${\rm sk}({\mathcal P})$ denote the $1$-skeleton of an convex polytope ${\mathcal P}$. Let $C$ be a clique (=complete subgraph) of ${\rm sk}({\mathcal P})$ and ${\rm conv}(C)$ the convex hull of the vertices of ${\mathcal P}$ belonging to $C$. In general, ${\rm conv}(C)$ may not be a face of ${\mathcal P}$. It will be proved that ${\rm conv}(C)$ is a face of ${\mathcal P}$ if ${\mathcal P}$ is either the order polytope ${\mathcal O}(P)$ of a finite partially ordered set $P$ or the stable set polytope ${\rm Stab}(G)$ of a finite simple graph $G$. In other words, when ${\mathcal P}$ is either ${\mathcal O}(P)$ or ${\rm Stab}(G)$, the simplicial complex consisting of simplices which are faces of ${\mathcal P}$ is the clique complex of ${\rm sk}({\mathcal P})$.