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On the Hyperbolicity Constant in Graph Minors

  • Walter Carballosa
  • José M. Rodríguez
  • Omar Rosario
  • José M. Sigarreta
Original Paper

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.

Keywords

Graph minor Edge contraction Deleted edge Hyperbolic graph Geodesics 

Mathematics Subject Classification

Primary 05C75 Secondary 05C63 05A20 

Notes

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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Copyright information

© Iranian Mathematical Society 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsFlorida International UniversityMiamiUSA
  2. 2.Department of MathematicsMiami Dade CollegeMiamiUSA
  3. 3.Department of MathematicsCarlos III University of MadridLeganésSpain
  4. 4.Faculty of MathematicsAutonomous University of GuerreroAcalpulco Gro.Mexico

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