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Acta Informatica

, Volume 35, Issue 12, pp 1075–1096 | Cite as

Axioms for generalized graphs, illustrated by a Cantor–Bernstein proposition

  • Joost Engelfriet
  • Tjalling Gelsema

Abstract.

The notion of a graph type \(\mathcal{T}\) is introduced by a collection of axioms. A graph of type \(\mathcal{T}\) (or \(\mathcal{T}\)-graph) is defined as a set of edges, of which the structure is specified by \(\mathcal{T}\). From this, general notions of subgraph and isomorphism of \(\mathcal{T}\)-graphs are derived. A Cantor-Bernstein (CB) result for \(\mathcal{T}\)-graphs is presented as an illustration of a general proof for different types of graphs. By definition, a relation \(\mathcal{R}\) on \(\mathcal{T}\)-graphs satisfies the CB property if \(A \mathcal{R} B\) and \(B \mathcal{R} A\) imply that A and B are isomorphic. In general, the relation ‘isomorphic to a subgraph’ does not satisfy the CB property. However, requiring the subgraph to be disconnected from the remainder of the graph, a relation that satisfies the CB property is obtained. A similar result is shown for \(\mathcal{T}\)-graphs with multiple edges.

Keywords

Generalize Graph General Notion Multiple Edge Graph Type General Proof 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Joost Engelfriet
    • 1
  • Tjalling Gelsema
    • 1
  1. 1. Department of Computer Science, Leiden University, P.O. Box 9512, NL-2300 RA Leiden, The Netherlands (e-mail: {engelfri,gelsema}@wi.leidenuniv.nl) NL

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