Vertex connectivity of chordal graphs

TL;DR

This paper establishes an upper bound on the vertex connectivity of chordal* graphs using algebraic methods and constructs graphs achieving any connectivity within this bound.

Contribution

It introduces a novel algebraic approach to bound vertex connectivity in chordal* graphs and provides explicit constructions for graphs with prescribed connectivity.

Findings

Vertex connectivity of chordal* graphs is bounded by a function of the number of vertices.

An algebraic proof using syzygy theory establishes the bound.

Explicit constructions of graphs with any allowable connectivity are provided.

Abstract

Let $G$ be a finite graph and $\kappa(G)$ the vertex connectivity of $G$. A chordal graph $G$ is called chordal$^*$ if no vertex of $G$ is adjacent to all other vertices of $G$. Using the syzygy theory in commutative algebra, it is proved that every chordal$^*$ graph $G$ on $n$ vertices satisfies $\kappa(G) \leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$. Furthermore, given an integer $0 \leq \kappa \leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$, a chordal$^*$ graph $G$ on $n$ vertices satisfying $\kappa(G) = \kappa$ is constructed.