Skip to main content
SpringerLink
Log in

Dichotomy Theorems for Homomorphism Polynomials of Graph Classes

  • Conference paper
WALCOM: Algorithms and Computation (WALCOM 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8973))

Included in the following conference series:

  • 715 Accesses

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bulatov, A.A., Grohe, M.: The complexity of partition functions. Theor. Comput. Sci. 348(2-3), 148–186 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bürgisser, P.: Completeness and reduction in algebraic complexity theory, vol. 7. Springer (2000)

    Google Scholar 

  3. Cai, J., Chen, X., Lu, P.: Graph homomorphisms with complex values: A dichotomy theorem. SIAM J. Comput. 42(3), 924–1029 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chekuri, C., Rajaraman, A.: Conjunctive query containment revisited. Theor. Comput. Sci. 239(2), 211–229 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. Dalmau, V., Jonsson, P.: The complexity of counting homomorphisms seen from the other side. Theor. Comput. Sci. 329(1-3), 315–323 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  7. Diestel, R.: Graph Theory. Springer, GmbH & Company KG, Berlin and Heidelberg (2000)

    Google Scholar 

  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. Dyer, M.E., Greenhill, C.S.: The complexity of counting graph homomorphisms (extended abstract). In: SODA, pp. 246–255 (2000)

    Google Scholar 

  10. Freuder, E.C.: Complexity of k-tree structured constraint satisfaction problems. In: AAAI, pp. 4–9 (1990)

    Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  12. Goldberg, L.A., Jerrum, M.: Counting unlabelled subtrees of a tree is #p-complete. LMS J. Comput. Math. 3, 117–124 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  13. Grohe, M., Thurley, M.: Counting homomorphisms and partition functions. Model Theoretic Methods in Finite Combinatorics 558, 243–292 (2011)

    Article  MathSciNet  Google Scholar 

  14. Grohe, M.: The complexity of homomorphism and constraint satisfaction problems seen from the other side. J. ACM 54(1) (2007)

    Google Scholar 

  15. Hell, P., Nešetřil, J.: On the complexity of h-coloring. Journal of Combinatorial Theory, Series B 48(1), 92–110 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hell, P., Nešetřil, J.: Graphs and homomorphisms, vol. 28. Oxford University Press, Oxford (2004)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  18. Miller, G.L.: An additivity theorem for the genus of a graph. J. Comb. Theory, Ser. B 43(1), 25–47 (1987)

    Article  MATH  Google Scholar 

  19. Mitchell, S.L.: Linear algorithms to recognize outerplanar and maximal outerplanar graphs. Information Processing Letters 9(5), 229–232 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  21. Strassen, V.: Vermeidung von divisionen. Journal für die reine und angewandte Mathematik 264, 184–202 (1973)

    MATH  MathSciNet  Google Scholar 

  22. Valiant, L.G.: Completeness classes in algebra. In: STOC, pp. 249–261 (1979)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Saarland University, Saarbruecken, Germany

    Christian Engels

Authors
  1. Christian Engels
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of CSE, BUET, ECE Building, West Palasi, 1205, Dhaka, Bangladesh

    M. Sohel Rahman

  2. The Advanced Algorithms Research Laboratory, The University of Electro-Communications, Chofugaoka 1-5-1, 182-8585, Chofu, Tokyo, Japan

    Etsuji Tomita

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Engels, C. (2015). Dichotomy Theorems for Homomorphism Polynomials of Graph Classes. In: Rahman, M.S., Tomita, E. (eds) WALCOM: Algorithms and Computation. WALCOM 2015. Lecture Notes in Computer Science, vol 8973. Springer, Cham. https://doi.org/10.1007/978-3-319-15612-5_25

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-15612-5_25

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15611-8

  • Online ISBN: 978-3-319-15612-5

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation