On Connectivity of the Facet Graphs of Simplicial Complexes

TL;DR

This paper investigates the connectivity properties of facet graphs in simplicial complexes, establishing specific vertex-connectivity levels for various types and extending the analysis to cell complexes.

Contribution

It provides new theoretical results on the vertex connectivity of facet graphs for different classes of simplicial complexes and extends these findings to cell complexes.

Findings

Facet graphs of $d$-cycles are $(d+1)$-vertex-connected.

Facet graphs of $d$-hypertrees are $d$-vertex-connected.

Facet graphs of $d$-hypercuts are $(n-d-1)$-vertex-connected.

Abstract

The paper studies the connectivity properties of facet graphs of simplicial complexes of combinatorial interest. In particular, it is shown that the facet graphs of $d$-cycles, $d$-hypertrees and $d$-hypercuts are, respectively, $(d+1)$, $d$, and $(n-d-1)$-vertex-connected. It is also shown that the facet graph of a $d$-cycle cannot be split into more than $s$ connected components by removing at most $s$ vertices. In addition, the paper discusses various related issues, as well as an extension to cell-complexes.