Dichotomy Theorems for Homomorphism Polynomials of Graph Classes

  • Christian Engels
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8973)


In this paper, we will show dichotomy theorems for the computation of polynomials corresponding to evaluation of graph homomorphisms in Valiant’s model. We are given a fixed graph H and want to find all graphs, from some graph class, homomorphic to this H. These graphs will be encoded by a family of polynomials.

We give dichotomies for the polynomials for cycles, cliques, trees, outerplanar graphs, planar graphs and graphs of bounded genus.


Hamiltonian Cycle Constraint Satisfaction Problem Dichotomy Theorem Graph Property Graph Class 
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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  1. 1.
    Bulatov, A.A., Grohe, M.: The complexity of partition functions. Theor. Comput. Sci. 348(2-3), 148–186 (2005)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bürgisser, P.: Completeness and reduction in algebraic complexity theory, vol. 7. Springer (2000)Google Scholar
  3. 3.
    Cai, J., Chen, X., Lu, P.: Graph homomorphisms with complex values: A dichotomy theorem. SIAM J. Comput. 42(3), 924–1029 (2013)zbMATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Chekuri, C., Rajaraman, A.: Conjunctive query containment revisited. Theor. Comput. Sci. 239(2), 211–229 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Dalmau, V., Jonsson, P.: The complexity of counting homomorphisms seen from the other side. Theor. Comput. Sci. 329(1-3), 315–323 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    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
  7. 7.
    Diestel, R.: Graph Theory. Springer, GmbH & Company KG, Berlin and Heidelberg (2000)Google Scholar
  8. 8.
    Durand, A., Mahajan, M., Malod, G., de Rugy-Althere, N., Saurabh, N.: Homomorphism polynomials complete for VP. In: FSTTCS ( to appear, 2014)Google Scholar
  9. 9.
    Dyer, M.E., Greenhill, C.S.: The complexity of counting graph homomorphisms (extended abstract). In: SODA, pp. 246–255 (2000)Google Scholar
  10. 10.
    Freuder, E.C.: Complexity of k-tree structured constraint satisfaction problems. In: AAAI, pp. 4–9 (1990)Google Scholar
  11. 11.
    Goldberg, L.A., Grohe, M., Jerrum, M., Thurley, M.: A complexity dichotomy for partition functions with mixed signs. SIAM J. Comput. 39(7), 3336–3402 (2010)zbMATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Goldberg, L.A., Jerrum, M.: Counting unlabelled subtrees of a tree is #p-complete. LMS J. Comput. Math. 3, 117–124 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Grohe, M., Thurley, M.: Counting homomorphisms and partition functions. Model Theoretic Methods in Finite Combinatorics 558, 243–292 (2011)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1) (2007)Google Scholar
  15. 15.
    Hell, P., Nešetřil, J.: On the complexity of h-coloring. Journal of Combinatorial Theory, Series B 48(1), 92–110 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Hell, P., Nešetřil, J.: Graphs and homomorphisms, vol. 28. Oxford University Press, Oxford (2004)zbMATHCrossRefGoogle Scholar
  17. 17.
    Mahajan, M., Rao, B.V.R.: Small space analogues of valiant’s classes and the limitations of skew formulas. Computational Complexity 22(1), 1–38 (2013)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Miller, G.L.: An additivity theorem for the genus of a graph. J. Comb. Theory, Ser. B 43(1), 25–47 (1987)zbMATHCrossRefGoogle Scholar
  19. 19.
    Mitchell, S.L.: Linear algorithms to recognize outerplanar and maximal outerplanar graphs. Information Processing Letters 9(5), 229–232 (1979)zbMATHMathSciNetCrossRefGoogle Scholar
  20. 20.
    de Rugy-Altherre, N.: A dichotomy theorem for homomorphism polynomials. In: Rovan, B., Sassone, V., Widmayer, P. (eds.) MFCS 2012. LNCS, vol. 7464, pp. 308–322. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  21. 21.
    Strassen, V.: Vermeidung von divisionen. Journal für die reine und angewandte Mathematik 264, 184–202 (1973)zbMATHMathSciNetGoogle Scholar
  22. 22.
    Valiant, L.G.: Completeness classes in algebra. In: STOC, pp. 249–261 (1979)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Christian Engels
    • 1
  1. 1.Department of Computer ScienceSaarland UniversitySaarbrueckenGermany

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