Ordered Sets pp 155-171 | Cite as

Graphs and Homomorphisms

  • Bernd Schröder


Aside from ordered sets, the fixed point property has been investigated in other settings. On one hand, the fixed point property is most likely originated in topology. (See Exercise 6-1 for the topological fixed point property.) On the other hand, in any branch of mathematics in which the underlying structures have a natural type of morphism, we can define a fixed point property as “every endomorphism has a fixed point.” Hence, graphs are natural discrete structures for which to investigate the fixed point property. However, we must be careful: For graphs, the natural analogues of order-preserving maps would be the simplicial homomorphisms from Definition 6.1. Nonetheless, graph theorists prefer the notion of a (graph) homomorphism from Definition 6.10. It would be fruitless to argue whether simplicial homomorphisms or graph homomorphisms are “more natural.” Each notion is natural in certain settings: For example, simplicial homomorphisms are in one-to-one correspondence with the simplicial maps of the clique complex of a graph, as we will see in Exercise  9-19 when we investigate simplicial complexes, whereas we will see in this chapter that graph homomorphisms are a good framework to investigate natural graph theoretical topics, such as, for example, colorings.


Complete Graph Simplicial Complex Comparability Graph Point Property Fixed Point Property 
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.


  1. 14.
    Bandelt, H. J., & van de Vel, M. (1987). A fixed cube theorem for median graphs. Discrete Mathematics, 67, 129–137.MathSciNetCrossRefzbMATHGoogle Scholar
  2. 16.
    Bélanger, M. F., Constantin, J., & Fournier, G. (1994). Graphes et ordonnés démontables, propriété de la clique fixe. Discrete Mathematics, 130, 9–17.MathSciNetCrossRefGoogle Scholar
  3. 48.
    Constantin, J., & Fournier, G. (1985). Ordonnés escamotables et points fixes. Discrete Mathematics, 53, 21–33.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 101.
    Gallai, T. (1967). Transitiv orientierbare graphen. Acta Mathematica Hungarica, 18, 25–66.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 105.
    Ghouilà-Houri, A. (1962). Caractérisation des graphes nonorientés dont on peut orienter les arêtes de manière à obtenir le graphe d’une relation d’ordre. Comptes Rendus de l’Académie des Sciences Paris, 254, 1370–1371.zbMATHGoogle Scholar
  6. 108.
    Gilmore, P., & Hoffman, J. (1963). A characterization of comparability graphs and of interval graphs. Canadian Journal of Mathematics, 16, 539–548.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 135.
    Hell, P., & Nešetril, J. (2004). Graphs and homomorphisms. Oxford lecture series in mathematics and its applications (Vol. 28). Oxford: Oxford University Press.CrossRefGoogle Scholar
  8. 148.
    Jawhari, E., Misane, D., & Pouzet, M. (1986). Retracts: Graphs and ordered sets from the metric point of view. In I. Rival (Ed.), Combinatorics and ordered sets. Contemporary mathematics (Vol. 57, pp. 175–226). Providence: American Mathematical Society.Google Scholar
  9. 154.
    Kelly, D. (1985). Comparability graphs. In I. Rival (Ed.), Graphs and order (pp. 3–40). Dordrecht: D. Reidel.CrossRefGoogle Scholar
  10. 165.
    Kleitman, D. J., & Rothschild, B. L. (1970). The number of finite topologies. Proceedings of the American Mathematical Society, 25, 276–282.Google Scholar
  11. 166.
    Kleitman, D. J., & Rothschild, B. L. (1975). Asymptotic enumeration of partial orders on a finite set. Transactions of the American Mathematical Society, 205, 205–220.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 200.
    Maróti, M., & Zádori, L. (2012). Reflexive digraphs with near unanimity polymorphisms. Discrete Mathematics, 312, 2316–2328.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 218.
    Nowakowski, R., & Rival, I. (1979). A fixed edge theorem for graphs with loops. Journal of Graph Theory, 3, 339–350.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 222.
    Polat, N. (1995). Retract-collapsible graphs and invariant subgraph properties. Journal of Graph Theory, 19, 25–44.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 223.
    Poston, T. (1971). Fuzzy Geometry. Ph.D. thesis, University of Warwick.Google Scholar
  16. 233.
    Quilliot, A. (1983). Homomorphismes, points fixes, rétractions et jeux de poursuite dans les graphes, les ensembles ordonnés et les espaces métriques. Thèse de doctorat d’état, Univ. Paris VI.Google Scholar
  17. 235.
    Quilliot, A. (1985). On the Helly property working as a compactness criterion for graphs. Journal of Combinatorial Theory (A), 40, 186–193.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 286.
    Schröder, B. (2015). Homomorphic constraint satisfaction problem solver. Google Scholar
  19. 301.
    Stong, R. E. (1966). Finite topological spaces. Transactions of the American Mathematical Society, 123, 325–340.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 304.
    Szymik, M. (2015). Homotopies and the universal fixed point. Order, 32, 301–311.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing 2016

Authors and Affiliations

  • Bernd Schröder
    • 1
  1. 1.Department of MathematicsUniversity of Southern MississippiHattiesburgUSA

Personalised recommendations