c Birkh¨auser Verlag, Basel, 2001
Annals of Combinatorics 5 (2001) 61-69
Annals of Combinatorics
0218-0006/01/010061-9$1.50+0.20/0
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
[email protected] Received February 26, 2000 AMS Subject Classification: 05C12, 05C05, 05C10, 05C75 Abstract. The hyperbolicity δ∗ ≥ 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 δ∗ = 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, δ-hyperbolic
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The author thanks the Graduiertenkolleg Strukturbildungsprozesse for its support. The author thanks the Swedish National Research Council (NFR)–grant# M12342-300–for its support, and also both FSPM-Strukturbildungsprozesse, University of Bielefeld and IFS, Massey University, for hosting him during part of this work.
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