Connectivity of cubical polytopes

TL;DR

This paper investigates the connectivity properties of graphs derived from cubical polytopes, establishing bounds on their connectivity and characterizing minimal separators for dimensions four and higher.

Contribution

It proves that the graph of any cubical d-polytope with minimum degree δ is min{δ, 2d-2}-connected and characterizes small separators as neighbors of a vertex for dimensions ≥ 4.

Findings

Graph connectivity is at least min{δ, 2d-2}.

Small minimal separators are exactly neighbors of a vertex.

Removing such separators disconnects the graph into two components.

Abstract

A cubical polytope is a polytope with all its facets being combinatorially equivalent to cubes. We deal with the connectivity of the graphs of cubical polytopes. We first establish that, for any $d\ge 3$, the graph of a cubical $d$-polytope with minimum degree $\delta$ is $\min\{\delta,2d-2\}$-connected. Second, we show, for any $d\ge 4$, that every minimum separator of cardinality at most $2d-3$ in such a graph consists of all the neighbours of some vertex and that removing the vertices of the separator from the graph leaves exactly two components, with one of them being the vertex itself.