Helly-gap of a graph and vertex eccentricities

Abstract

A new metric parameter for a graph, Helly-gap, is introduced. A graph G is called α-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 α. The minimum α for which a graph G is α-weakly-Helly is called the Helly-gap of G and denoted by α(G). The Helly-gap of a graph G is characterized by distances in the injective hull 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 (α(G)=0), as well as for chordal graphs (α(G)≤1), distance-hereditary graphs (α(G)≤1) and δ-hyperbolic graphs (α(G)≤2δ), to all graphs, parameterized by their Helly-gap α(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 αi-metric.