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A Category of “Undirected Graphs”

A Tribute to Hartmut Ehrig

  • Chapter
  • First Online:
Graph Transformation, Specifications, and Nets

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10800))

Abstract

In this paper, a category of undirected graphs is introduced where the morphisms are chosen in the style of mathematical graph theory rather than as algebraic structures as is more usual in the area of graph transformation.

A representative function, \({\omega }\), within this category is presented. Its inverse, \({\omega }^{-1}\), is defined in terms of a graph grammar, \({\varepsilon }\).

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Notes

  1. 1.

    The codomain \(2^{V'}\) of f need not be \(2^V\), and its edge set \(E'\) need not have the same structure as E. Therefore, elements of the codomain are denoted with a prime.

  2. 2.

    We use suffix notation to denote the application of set-valued operators and functions.

  3. 3.

    This is a common terminology, but unfortunately such “closed neighborhoods” are not “closed”. The intersection of closed sets must be closed, but it easy to show that this is seldom true with “closed neighborhoods”.

  4. 4.

    We modify the usual definition of monotonicity to read: \(X \subseteq Y\) implies \(X.f \subseteq Y.f\), provided .

  5. 5.

    This procedure has been quite effective reducing large graphs \(| V | \ge 1,000\), with at worst 6 iterative sweeps of V.

  6. 6.

    A graph, \(K_n\) is complete if all n nodes are mutually connected by an edge.

  7. 7.

    Because extreme points are simplicial (neighborhood is a clique), and because every chordal graph must have at least two extreme points [8, 9], every chordal graph can be so generated.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Virginia, Charlottesville, USA

    John L. Pfaltz

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  1. John L. Pfaltz
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Correspondence to John L. Pfaltz .

Editor information

Editors and Affiliations

  1. Department of Informatics, University of Leicester, Leicester, United Kingdom

    Reiko Heckel

  2. Fachbereich Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Gabriele Taentzer

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Pfaltz, J.L. (2018). A Category of “Undirected Graphs”. In: Heckel, R., Taentzer, G. (eds) Graph Transformation, Specifications, and Nets. Lecture Notes in Computer Science(), vol 10800. Springer, Cham. https://doi.org/10.1007/978-3-319-75396-6_12

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  • DOI: https://doi.org/10.1007/978-3-319-75396-6_12

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  • Online ISBN: 978-3-319-75396-6

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