Hyperbolicity and Chordality of a Graph

Abstract

Let $G$ be a connected graph with the usual shortest-path metric $d$. The graph $G$ is $\delta$-hyperbolic provided for any vertices $x,y,u,v$ in it, the two larger of the three sums $d(u,v)+d(x,y),d(u,x)+d(v,y)$ and $d(u,y)+d(v,x)$ differ by at most $2\delta.$ The graph $G$ is $k$-chordal provided it has no induced cycle of length greater than $k.$ Brinkmann, Koolen and Moulton find that every $3$-chordal graph is $1$-hyperbolic and that graph is not $\frac{1}{2}$-hyperbolic if and only if it contains one of two special graphs as an isometric subgraph. For every $k\geq 4,$ we show that a $k$-chordal graph must be $\frac{\lfloor\frac{k}{2}\rfloor}{2}$-hyperbolic and there does exist a $k$-chordal graph which is not $\frac{\lfloor \frac{k-2}{2}\rfloor}{2}$-hyperbolic. Moreover, we prove that a $5$-chordal graph is $\frac{1}{2}$-hyperbolic if and only if it does not contain any of a list of five special graphs as an isometric subgraph.

On the hyperbolicity of bipartite graphs and intersection graphs

“…Relating the structural properties of graphs with hyperbolicity can be useful in this context, and it has become a growing line of research (e.g., see [12,15,24,31]). Indeed, we argue that one can obtain from such relations a comprehensive overview of the reasons why some complex networks are hyperbolic and some others are not.…”

Chordality Properties and Hyperbolicity on Graphs

“…In [7], the authors prove that chordal graphs are hyperbolic giving an upper bound for the hyperbolicity constant. In [40], Wu and Zhang extend this result for a generalized version of chordality. They prove that k-chordal graphs are hyperbolic where a graph is k-chordal if every induced cycle has at most k edges.…”

Metric tree‐like structures in real‐world networks: an empirical study

“…‐Hyperbolicity measures the local deviation of a metric from a tree metric; a graph metric is a tree metric if and only if it has hyperbolicity . Note that chordal graphs, mentioned in Section 4, have hyperbolicity at most , while ‐chordal graphs have hyperbolicity at most .…”