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 $\alpha (G)$. 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 δ-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.

引入了图表的新度量参数Helly-gap。如果当每个圆盘的半径增加一个附加值α时，如果G中成对相交圆盘的任何系统具有非空公共交点，则图G称为α-弱螺旋。图G为α -weakly-Helly的最小α称为G的Helly-gap，用$\alpha （G）$。图G的Helly间隙的特征是内射壳的距离$\mathcal{H}（G）$，这是一个（唯一的）最小Helly图，其中包含G作为等距子图。此表征用作对Helly图已知的许多与偏心率相关的结果进行归纳的工具（$\alpha （G）=0$），以及和弦图（$\alpha （G）\le \mathrm{1个}$），距离遗传图（$\alpha （G）\le \mathrm{1个}$）和δ-双曲图（$\alpha （G）\le \mathrm{2个}\delta$），所有图表均由其Helly-gap参数化 $\alpha （G）$。还显示了其他几种具有有界Helly间隙的图类，包括无AT图和具有有界树长，有界弦或有界的图${\alpha }_{\mathrm{一世}}$-metric。