Helly-gap of a graph and vertex eccentricities

• Published in: arXiv.org
• Main Authors: Dragan, Feodor F, Guarnera, Heather M
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 05.05.2020
• Subjects: Disks; Graph theory; Graphs
• ISSN: 2331-8422

A new metric parameter for a graph, Helly-gap, is introduced. A graph \(G\) is called \(\alpha\)-weakly-Helly if any system of pairwise intersecting disks in \(G\) has a nonempty common intersection when the radius of each disk is increased by an additive value \(\alpha\). The minimum \(\alpha\) for which a graph \(G\) is \(\alpha\)-weakly-Helly is called the Helly-gap of \(G\) and denoted by \(\alpha(G)\). The Helly-gap of a graph \(G\) is characterized by distances in the injective hull \(\mathcal{H}(G)\), which is a (unique) minimal Helly graph which contains \(G\) as an isometric subgraph. This characterization is used as a tool to generalize many eccentricity related results known for Helly graphs (\(\alpha(G)=0\)), as well as for chordal graphs (\(\alpha(G)\le 1\)), distance-hereditary graphs (\(\alpha(G)\le 1\)) and \(\delta\)-hyperbolic graphs (\(\alpha(G)\le 2\delta\)), to all graphs, parameterized by their Helly-gap \(\alpha(G)\). Several additional graph classes are shown to have a bounded Helly-gap, including AT-free graphs and graphs with bounded tree-length, bounded chordality or bounded \(\alpha_i\)-metric.