Obstructions to a small hyperbolicity in Helly graphs

TL;DR

This paper investigates the structural properties of Helly graphs that influence their hyperbolicity, identifying specific subgraphs that obstruct small hyperbolicity in these graphs.

Contribution

It identifies three isometric subgraphs of the King-grid as structural obstructions to small hyperbolicity in Helly graphs.

Findings

Three isometric subgraphs obstruct small hyperbolicity.

Structural properties of Helly graphs influence hyperbolicity.

Obstructions are identified within the King-grid subgraphs.

Abstract

It is known that for every graph $G$ there exists the smallest Helly graph $\cal H(G)$ into which $G$ isometrically embeds ($\cal H(G)$ is called the injective hull of $G$) such that the hyperbolicity of $\cal H(G)$ is equal to the hyperbolicity of $G$. Motivated by this, we investigate structural properties of Helly graphs that govern their hyperbolicity and identify three isometric subgraphs of the King-grid as structural obstructions to a small hyperbolicity in Helly graphs.