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A Category of “Undirected Graphs”

A Tribute to Hartmut Ehrig
  • John L. PfaltzEmail author
Chapter
Part of the Lecture Notes in Computer Science book series (LNCS, 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, \({\varepsilon }\).

References

  1. 1.
    Arbib, M., Manes, E.: Arrows, Structures, and Functors: The Categorical Imperative. Academic Press, New York (1975)zbMATHGoogle Scholar
  2. 2.
    Castellini, G.: Categorical Closure Operators. Birkhauser, Boston (2003)CrossRefzbMATHGoogle Scholar
  3. 3.
    Chvátal, V.: Antimatroids, betweenness, convexity. In: László, W.C., Vygen, J. (eds.) Research Trends in Combinatorial Optimization, pp. 57–64. Springer, Heidelberg (2009).  https://doi.org/10.1007/978-3-540-76796-1_3 CrossRefGoogle Scholar
  4. 4.
    Edelman, P.H.: Abstract convexity and meet-distributive lattices. In: Combinatorics and Ordered Sets, Arcata, CA, pp. 127–150 (1986)Google Scholar
  5. 5.
    Edelman, P.H., Jamison, R.E.: The theory of convex geometries. Geom. Dedicata 19(3), 247–270 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ehrig, H., Pfender, M., Schneider, H.J.: Graph grammars: an algebraic approach. In: IEEE Conference on SWAT (1973)Google Scholar
  7. 7.
    Engle, K.: Sperner theory. In: Hazewinkle, M. (ed.) Encyclopedia of Mathematics. Springer, Heidelberg (2001)Google Scholar
  8. 8.
    Farber, M., Jamison, R.E.: Convexity in graphs and hypergraphs. SIAM J. Algebra Discrete Methods 7(3), 433–444 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)zbMATHGoogle Scholar
  10. 10.
    MacLane, S.: Categories for the Working Mathematician, 2nd edn. Springer, New York (1998).  https://doi.org/10.1007/978-1-4612-9839-7 zbMATHGoogle Scholar
  11. 11.
    Ore, O.: Mappings of closure relations. Ann. Math. 47(1), 56–72 (1946)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Pfaltz, J., Šlapal, J.: Transformations of discrete closure systems. Acta Math. Hung. 138(4), 386–405 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Pfaltz, J.L.: Neighborhood expansion grammars. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) TAGT 1998. LNCS, vol. 1764, pp. 30–44. Springer, Heidelberg (2000).  https://doi.org/10.1007/978-3-540-46464-8_3 CrossRefGoogle Scholar
  14. 14.
    Pfaltz, J.L.: Finding the mule in the network. In: Alhajj, R., Werner, B. (eds.) International Conference on Advances in Social Network Analysis and Mining, ASONAM 2012, Istanbul, Turkey, pp. 667–672, August 2012Google Scholar
  15. 15.
    Pfaltz, J.L.: Mathematical continuity in dynamic social networks. Soc. Netw. Anal. Min. (SNAM) 3(4), 863–872 (2013)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Pfaltz, J.L.: The irreducible spine(s) of undirected networks. In: Lin, X., Manolopoulos, Y., Srivastava, D., Huang, G. (eds.) WISE 2013. LNCS, vol. 8181, pp. 104–117. Springer, Heidelberg (2013).  https://doi.org/10.1007/978-3-642-41154-0_8 CrossRefGoogle Scholar
  17. 17.
    Pfaltz, J.L.: Computational processes that appear to model human memory. In: Figueiredo, D., Martín-Vide, C., Pratas, D., Vega-Rodríguez, M.A. (eds.) AlCoB 2017. LNCS, vol. 10252, pp. 85–99. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-58163-7_6 CrossRefGoogle Scholar
  18. 18.
    Pierce, B.C.: Basic Category Theory for Computer Scientists. MIT Press, Cambridge (1991)zbMATHGoogle Scholar
  19. 19.
    Rozenberg, G. (ed.): The Handbook of Graph Grammars. World Scientific, Singapore (1997)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of VirginiaCharlottesvilleUSA

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