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arXiv · 2502.07320

Vertex connectivity of chordal graphs

Tài Huy Hà·Takayuki Hibi
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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.

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Tài Huy Hà, Takayuki Hibi. 2025-02-11. Vertex connectivity of chordal graphs. https://arxiv.org/abs/2502.07320

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