Vertex connectivity of chordal graphs

TL;DR

The paper studies the vertex connectivity $kappa(G)$ of chordal^* graphs on $n$ vertices (graphs with no universal vertex) using syzygy theory of monomial ideals. It proves a sharp bound $0 <= kappa(G) <= (n-1) - ceil(2*sqrt(n) - 2)$ and shows $kappa(G) + tau_max(G^c) <= n-1$, with every value in that range realizable by some chordal^* graph. The construction relates kappa(G) to the projective dimension of $S/I(G^c)$ and to a dual graph $G^c$ via the MAXMIN framework, yielding a realizability result for all kappa and a corollary $a(G) <= (n-1) - ceil(2*sqrt(n) - 2)$ for the algebraic connectivity. Section 2 then classifies bound-attaining graphs, showing non-uniqueness in general but unique realizations when $n$ is a perfect square, with $G = H^c$ linked to a unique $H$ having $tau_max(H) = ceil(2*sqrt(n) - 2)$ via the MAXMIN construction.

Abstract

Let $G$ be a finite graph and $κ(G)$ 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 $κ(G) \leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$. Furthermore, given an integer $0 \leq κ\leq (n - 1) - \lceil2\sqrt{n}-2\,\rceil$, a chordal$^*$ graph $G$ on $n$ vertices satisfying $κ(G) = κ$ is constructed.

Figures (1)

• Figure 1: Chordal$^*$ graphs with $6$ vertices

Theorems & Definitions (7)

• Theorem 1.1
• proof
• Example 1.2
• Corollary 1.3
• Example 2.1
• Theorem 2.2
• proof