Addendum to Vertex adjacencies in the set covering polyhedron

TL;DR

This paper explores how truncating an up-monotone polyhedron with the unit hypercube affects its vertices, revealing their relationships and properties, especially when the original vertices are binary.

Contribution

It characterizes the vertices of the truncated polytope in relation to the original, proving integrality and subgraph properties, and applies these results to clarify a previous claim.

Findings

Vertices of the truncated polytope are characterized in terms of original vertices.

The vertices of the original polyhedron are shown to be integral.

The 1-skeleton of the original polyhedron is an induced subgraph of the truncated polytope's 1-skeleton.

Abstract

We study the relationship between the vertices of an up-monotone polyhedron $R$ and those of the polytope $P$ obtained by truncating $R$ with the unit hypercube. When $R$ has binary vertices, we characterize the vertices of $P$ in terms of the vertices of $R$, show their integrality, and prove that the 1-skeleton of $R$ is an induced subgraph of the 1-skeleton of $P$. We conclude by applying our findings to settle a claim in the original paper.