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On the Hyperbolicity of Chordal Graphs
Gunnar Brinkmann1, Jack H. Koolen1, and V. Moulton2
1FSPM-Strukturbildungsprozesse, University of Bielefeld, D-33501 Bielefeld, Germany
{gunnar, jkoolen}@mathematik.uni-bielefeld.de
2Physics and Mathematics Department (FMI), Mid Sweden University, Sundsvall, S 851-70, Sweden
Annals of Combinatorics 5(1) p.61-69 March, 2001
AMS Subject Classification: 05C12, 05C05, 05C10, 05C75
The hyperbolicity $ \delta^* \ge 0$ of a metric space in Gromov's sense can be viewed as a measure of how "tree-like" the space is, since those spaces for which $ \delta^* = 0$ holds are precisely the set of (metric) trees. Here, we show that any chordal graph equipped with the usual graph metric is in this sense reasonably tree-like. In particular, we prove that the hyperbolicity of any chordal graph is bounded, and is at most two. Moreover, we characterize those chordal graphs with hyperbolicity one.
Keywords: chordal graphs, block graphs, bridged graphs, four-point condition, hyperbolicity, $ \delta$-hyperbolic


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