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On the Hyperbolicity Constant in Graph Minors

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Abstract

A graph H is a minor of a graph G if a graph isomorphic to H can be obtained from G by contracting some edges, deleting some edges, and deleting some isolated vertices. The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it. One of the main aims in this work is to obtain quantitative information about the distortion of the hyperbolicity constant of the graph \(\,G/e\) obtained from the (simple or non-simple) graph G by contracting an arbitrary edge e from it. We prove that H is hyperbolic if and only if G is hyperbolic, for many minors H of G, even if H is obtained from G by contracting and/or deleting infinitely many edges.

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Acknowledgements

This work was supported in part by two grants from Ministerio de Economía y Competititvidad (MTM2013-46374-P and MTM2015-69323-REDT), Spain, and a Grant from CONACYT (FOMIX-CONACyT-UAGro 249818), México. Additionally, we would like to thank the referee for him/her careful reading of the manuscript and several useful comments which have helped us to improve the presentation of this paper.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Florida International University, 11200 SW 8th Street, Miami, FL, 33199, USA

    Walter Carballosa

  2. Department of Mathematics, Miami Dade College, 300 NE Second Ave., Miami, FL, 33132, USA

    Walter Carballosa

  3. Department of Mathematics, Carlos III University of Madrid, Avenida de la Universidad 30, 28911, Leganés, Madrid, Spain

    José M. Rodríguez

  4. Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame No.54 Col. Garita, 39650, Acalpulco Gro., Mexico

    Omar Rosario & José M. Sigarreta

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  1. Walter Carballosa
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Correspondence to José M. Rodríguez.

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Carballosa, W., Rodríguez, J.M., Rosario, O. et al. On the Hyperbolicity Constant in Graph Minors. Bull. Iran. Math. Soc. 44, 481–503 (2018). https://doi.org/10.1007/s41980-018-0032-y

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