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Strong Isoperimetric Inequalities and Combinatorial Curvatures on Multiply Connected Planar Graphs

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Abstract

The main focus of this paper is on hyperbolic properties of multiply connected planar graphs (planar graphs with multiple ends), and in the course we study some problematic phenomena of planar graphs caused by the existence of multiple (or sometimes infinite) ends. Specifically, in the first part of the paper we examine strong isoperimetric inequalities on a multiply connected planar graph G and its dual graph \(G^*\), and prove that G satisfies a strong isoperimetric inequality if and only if \(G^*\) has the same property, provided that G is either normal or finitely connected and we choose an appropriate notion for strong isoperimetric inequalities. In the second part we study a planar graph G on which negative curvatures uniformly dominate positive curvatures, and give a criterion that guarantees a strong isoperimetric inequality on G. Our criterion is useful in that it can be applied to a graph containing a long and slim subgraph with nonnegative combinatorial curvatures.

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Acknowledgments

The authors deeply appreciate the anonymous referees for their helpful comments and suggestions, which helped to improve the presentation of this paper in many aspects. This paper was revised while B. Oh was visiting Korea Institute for Advanced Study (KIAS) in 2016. He thanks KIAS for warm hospitality. B. Oh was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (2010-0004113).

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Authors and Affiliations

  1. Department of Mathematics Education, Hanyang University, 222 Wangsimni-ro, Seongdong-gu, Seoul, 04763, Korea

    Byung-Geun Oh

  2. Department of Mathematics, Soongsil University, 369 Sangdo-Ro, Dongjak-Gu, Seoul, 06978, Korea

    Jeehyeon Seo

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  1. Byung-Geun Oh
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  2. Jeehyeon Seo
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Correspondence to Byung-Geun Oh.

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Oh, BG., Seo, J. Strong Isoperimetric Inequalities and Combinatorial Curvatures on Multiply Connected Planar Graphs. Discrete Comput Geom 56, 558–591 (2016). https://doi.org/10.1007/s00454-016-9805-3

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  • DOI: https://doi.org/10.1007/s00454-016-9805-3

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