Counting Homomorphisms to Square-Free Graphs, Modulo 2

  • Andreas Göbel
  • Leslie Ann Goldberg
  • David Richerby
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


We study the problem \(\oplus \textsc {HomsTo}{H}\) of counting, modulo 2, the homomorphisms from an input graph to a fixed undirected graph H. A characteristic feature of modular counting is that cancellations make wider classes of instances tractable than is the case for exact (non-modular) counting, so subtle dichotomy theorems can arise. We show the following dichotomy: for any H that contains no 4-cycles, \(\oplus \textsc {HomsTo}{H}\) is either in polynomial time or is \(\oplus \mathrm {P}\)-complete. This partially confirms a conjecture of Faben and Jerrum that was previously only known to hold for trees and for a restricted class of tree-width-2 graphs called cactus graphs. We confirm the conjecture for a rich class of graphs including graphs of unbounded tree-width. In particular, we focus on square-free graphs, which are graphs without 4-cycles. These graphs arise frequently in combinatorics, for example in connection with the strong perfect graph theorem and in certain graph algorithms. Previous dichotomy theorems required the graph to be tree-like so that tree-like decompositions could be exploited in the proof. We prove the conjecture for a much richer class of graphs by adopting a much more general approach.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arends, F., Ouaknine, J., Wampler, C.W.: On searching for small Kochen-Specker vector systems. In: Kolman, P., Kratochvíl, J. (eds.) WG 2011. LNCS, vol. 6986, pp. 23–34. Springer, Heidelberg (2011) CrossRefGoogle Scholar
  2. 2.
    Bulatov, A.A., Grohe, M.: The complexity of partition functions. Theor. Comput. Sci. 348(2–3), 148–186 (2005)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Cai, J.-Y., Lu, P.: Holographic algorithms: From art to science. J. Comput. Syst. Sci. 77(1), 41–61 (2011)MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Conforti, M., Cornuéjols, G., Vušković, K.: Square-free perfect graphs. J. Combin. Theory Ser. B 90(2), 257–307 (2004)MATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Dyer, M.E., Greenhill, C.S.: The complexity of counting graph homomorphisms. Random Struct. Algorithms 17(3–4), 260–289 (2000)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Faben, J., Jerrum, M.: The complexity of parity graph homomorphism: an initial investigation. Theor. Comput. 11, 35–57 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Göbel, A., Goldberg, L.A., Richerby, D.: The complexity of counting homomorphisms to cactus graphs modulo 2. ACM T. Comput. Theory, 6(4), article 17 (2014)Google Scholar
  8. 8.
    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)MATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Goldberg, L.A., Gysel, R., Lapinskas, J.: Approximately counting locally-optimal structures. CoRR, abs/1411.6829 (2014)Google Scholar
  10. 10.
    Goldschlager, L.M., Parberry, I.: On the construction of parallel computers from various bases of Boolean functions. Theor. Comput. Sci. 43, 43–58 (1986)MATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Hell, P., Nešetřil, J.: On the complexity of \(H\)-coloring. J. Comb. Theory, Ser. B 48(1), 92–110 (1990)MATHCrossRefGoogle Scholar
  12. 12.
    Lovász, L.: Operations with structures. Acta Math. Acad. Sci. Hungar. 18(3–4), 321–328 (1967)MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Papadimitriou, C.H., Zachos, S.: Two remarks on the power of counting. In: Cremers, A.B., Kriegel, H.-P. (eds.) Theoretical Computer Science. LNCS, vol. 145, pp. 269–275. Springer, Heidelberg (1982) CrossRefGoogle Scholar
  14. 14.
    Schaefer, T.J.: The complexity of satisfiability problems. In: Proc. 10th Annual ACM Symposium on Theory of Computing (STOC 1978), pp. 216–226. ACM Press (1978)Google Scholar
  15. 15.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20(5), 865–877 (1991)MATHMathSciNetCrossRefGoogle Scholar
  16. 16.
    Valiant, L.G.: Accidental algorithms. In: Proc. 47th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2006), pp. 509–517. IEEE (2006)Google Scholar
  17. 17.
    Xia, M., Zhang, P., Zhao, W.: Computational complexity of counting problems on \(3\)-regular planar graphs. Theoret. Comput. Sci. 384(1), 111–125 (2007)MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • Andreas Göbel
    • 1
  • Leslie Ann Goldberg
    • 1
  • David Richerby
    • 1
  1. 1.Department of Computer ScienceUniversity of OxfordOxfordUK

Personalised recommendations