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The Grid Property and Product-Like Hypergraphs

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Abstract

Equivalence relations on the edge set of a hypergraph that satisfy the “grid property” (a certain restrictive condition on diagonal-free grids that can be seen as a generalization of the more familiar “square property” on graphs) play a crucial role in the theory of Cartesian hypergraph products. In particular, every convex relation with the grid property induces a factorization w.r.t. the Cartesian product. In the class of graphs, even non-convex relations with the square property provide rich structural information on local isomorphisms, local product structures, and product structures of quotient graphs. Here, we examine the grid property in its own right. Vertex partitions derived from these equivalence classes of the edges give rise to equivalence relations on the vertex set. This in turn determines quotient graphs that have non-trivial product structures.

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Authors and Affiliations

  1. Bioinformatics Group, Department of Computer Science, and Interdisciplinary Center for Bioinformatics, University of Leipzig, Härtelstraße 16-18, 04107, Leipzig, Germany

    Lydia Ostermeier & Peter F. Stadler

  2. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

    Lydia Ostermeier & Peter F. Stadler

  3. RNomics Group, Fraunhofer Institut für Zelltherapie und Immunologie-IZI, Perlickstraße 1, 04103, Leipzig, Germany

    Peter F. Stadler

  4. Department of Theoretical Chemistry, University of Vienna, Währingerstraße 17, 1090, Wien, Austria

    Peter F. Stadler

  5. Center for non-coding RNA in Technology and Health, University of Copenhagen, Grønnegårdsvej 3, 1870, Frederiksberg C, Denmark

    Peter F. Stadler

  6. Santa Fe Institute, 1399 Hyde Park Rd., Santa Fe, NM, 87501, USA

    Peter F. Stadler

Authors
  1. Lydia Ostermeier
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  2. Peter F. Stadler
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Correspondence to Lydia Ostermeier.

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Ostermeier, L., Stadler, P.F. The Grid Property and Product-Like Hypergraphs. Graphs and Combinatorics 31, 757–770 (2015). https://doi.org/10.1007/s00373-013-1392-y

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  • DOI: https://doi.org/10.1007/s00373-013-1392-y

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