Locally Constrained Homomorphisms on Graphs of Bounded Treewidth and Bounded Degree

  • Steven Chaplick
  • Jiří Fiala
  • Pim van ’t Hof
  • Daniël Paulusma
  • Marek Tesař
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8070)


A homomorphism from a graph G to a graph H is locally bijective, surjective, or injective if its restriction to the neighborhood of every vertex of G is bijective, surjective, or injective, respectively. We prove that the problems of testing whether a given graph G allows a homomorphism to a given graph H that is locally bijective, surjective, or injective, respectively, are NP-complete, even when G has pathwidth at most 5, 4 or 2, respectively, or when both G and H have maximum degree 3. We complement these hardness results by showing that the three problems are polynomial-time solvable if G has bounded treewidth and in addition G or H has bounded maximum degree.


Polynomial Time Maximum Degree Constraint Satisfaction Problem Tree Decomposition Surjective Homomorphism 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angluin, D.: Local and global properties in networks of processors. In: Proc. STOC 1980, pp. 82–93 (1980)Google Scholar
  2. 2.
    Angluin, D., Gardiner, A.: Finite common coverings of pairs of regular graphs. J. Comb. Theory Ser. B 30, 184–187 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Biggs, N.: Algebraic Graph Theory. Cambridge University Press (1974)Google Scholar
  4. 4.
    Biggs, N.: Constructing 5-arc transitive cubic graphs. J. London Math. Society II 26, 193–200 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L.: The classification of coverings of processor networks. J. Par. Distrib. Comp. 6, 166–182 (1989)CrossRefGoogle Scholar
  6. 6.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comp. 25(6), 1305–1317 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Chalopin, J., Métivier, Y., Zielonka, W.: Election, naming and cellular edge local computations. In: Ehrig, H., Engels, G., Parisi-Presicce, F., Rozenberg, G. (eds.) ICGT 2004. LNCS, vol. 3256, pp. 242–256. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  8. 8.
    Dalmau, V., Kolaitis, P.G., Vardi, M.Y.: Constraint satisfaction, bounded treewidth, and finite-variable logics. In: Van Hentenryck, P. (ed.) CP 2002. LNCS, vol. 2470, pp. 310–326. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  9. 9.
    Everett, M.G., Borgatti, S.: Role coloring a graph. Mathematical Social Sciences 21, 183–188 (1991)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Fiala, J., Kratochvíl, J.: Locally constrained graph homomorphisms – Structure, complexity, and applications. Comp. Sci. Review 2, 97–111 (2008)CrossRefGoogle Scholar
  11. 11.
    Fiala, J., Kratochvíl, J.: Partial covers of graphs. Disc. Math. Graph Theory 22, 89–99 (2002)zbMATHCrossRefGoogle Scholar
  12. 12.
    Fiala, J., Kratochvíl, J., Kloks, T.: Fixed-parameter complexity of λ-labelings. Discr. Appl. Math. 113, 59–72 (2001)zbMATHCrossRefGoogle Scholar
  13. 13.
    Fiala, J., Paulusma, D.: A complete complexity classification of the role assignment problem. Theor. Comp. Sci. 349, 67–81 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Fiala, J., Paulusma, D.: Comparing universal covers in polynomial time. Theory Comp. Syst. 46, 620–635 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Galluccio, A., Hell, P., Nešetřil, J.: The complexity of H-colouring of bounded degree graphs. Discr. Math. 222, 101–109 (2000)zbMATHCrossRefGoogle Scholar
  16. 16.
    Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-completeness. W. H. Freeman & Co., New York (1979)Google Scholar
  17. 17.
    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54 (2007)Google Scholar
  18. 18.
    Gurski, F., Wanke, E.: The tree-width of clique-width bounded graphs without K n,n. In: Brandes, U., Wagner, D. (eds.) WG 2000. LNCS, vol. 1928, pp. 196–205. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  19. 19.
    Hell, P., Nešetřil, J.: On the complexity of H-colouring. J. Comb. Theory Ser. B 48, 92–110 (1990)zbMATHCrossRefGoogle Scholar
  20. 20.
    Hell, P., Nešetřil, J.: Graphs and Homomorphisms. Oxford University Press (2004)Google Scholar
  21. 21.
    Heggernes, P., van ’t Hof, P., Paulusma, D.: Computing role assignments of proper interval graphs in polynomial time. J. Discr. Alg. 14, 173–188 (2012)zbMATHCrossRefGoogle Scholar
  22. 22.
    Kloks, T.: Treewidth, Computations and Approximations. LNCS, vol. 842. Springer (1994)Google Scholar
  23. 23.
    Kratochvíl, J., Křivánek, M.: On the computational complexity of codes in graphs. In: Koubek, V., Janiga, L., Chytil, M.P. (eds.) MFCS 1988. LNCS, vol. 324, pp. 396–404. Springer, Heidelberg (1988)CrossRefGoogle Scholar
  24. 24.
    Kratochvíl, J., Proskurowski, A., Telle, J.A.: Covering regular graphs. J. Comb. Theory Ser. B 71, 1–16 (1997)zbMATHCrossRefGoogle Scholar
  25. 25.
    Kristiansen, P., Telle, J.A.: Generalized H-coloring of graphs. In: Lee, D.T., Teng, S.-H. (eds.) ISAAC 2000. LNCS, vol. 1969, pp. 456–466. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  26. 26.
    Massey, W.S.: Algebraic Topology: An Introduction. Harcourt, Brace and World (1967)Google Scholar
  27. 27.
    Nešetřil, J.: Homomorphisms of derivative graphs. Discr. Math. 1, 257–268 (1971)zbMATHCrossRefGoogle Scholar
  28. 28.
    Pekeč, A., Roberts, F.S.: The role assignment model nearly fits most social networks. Mathematical Social Sciences 41, 275–293 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Roberts, F.S., Sheng, L.: How hard is it to determine if a graph has a 2-role assignment? Networks 37, 67–73 (2001)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Steven Chaplick
    • 1
  • Jiří Fiala
    • 1
  • Pim van ’t Hof
    • 2
  • Daniël Paulusma
    • 3
  • Marek Tesař
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPragueCzech Republic
  2. 2.Department of InformaticsUniversity of BergenNorway
  3. 3.School of Engineering and Computing SciencesDurham UniversityUK

Personalised recommendations