Skip to main content
SpringerLink
Log in

Linear Recurrence Relations for Graph Polynomials

  • Chapter
Pillars of Computer Science

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4800))

Abstract

A sequence of graphs G n is iteratively constructible if it can be built from an initial labeled graph by means of a repeated fixed succession of elementary operations involving addition of vertices and edges, deletion of edges, and relabelings. Let G n be a iteratively constructible sequence of graphs. In a recent paper, [27], M. Noy and A. Ribò have proven linear recurrences with polynomial coefficients for the Tutte polynomials T(G i , x,y) = T(G i ), i.e.

T(Gn + r) = p1(x,y) T(Gn + r − 1) + ... + p r (x,y) T(G n ).

We show that such linear recurrences hold much more generally for a wide class of graph polynomials (also of labeled or signed graphs), namely they hold for all the extended MSOL-definable graph polynomials. These include most graph and knot polynomials studied in the literature.

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 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
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. Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial: a new graph polynomial. Journal of Combinatorial Theory, Series B 92, 199–233 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Arratia, R., Bollobás, B., Sorkin, G.B.: A two-variable interlace polynomial. Combinatorica 24(4), 567–584 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  3. Biggs, N.: Algebraic Graph Theory, 2nd edn. Cambridge University Press, Cambridge (1993)

    Google Scholar 

  4. Biggs, N.L., Damerell, R.M., Sand, D.A.: Recursive families of graphs. J. Combin. Theory Ser. B 12, 123–131 (1972)

    Article  MATH  Google Scholar 

  5. Bollobás, B.: Modern Graph Theory. Springer, Heidelberg (1999)

    Google Scholar 

  6. Bollobás, B., Riordan, O.: A Tutte polynomial for coloured graphs. Combinatorics, Probability and Computing 8, 45–94 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chung, F.R.K., Graham, R.L.: On the cover polynomial of a digraph. Journal of Combinatorial Theory, Ser. B 65(2), 273–290 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  8. Courcelle, B.: A multivariate interlace polynomial (December 2006) (preprint)

    Google Scholar 

  9. Courcelle, B., Makowsky, J.A.: Fusion on relational structures and the verification of monadic second order properties. Mathematical Structures in Computer Science 12(2), 203–235 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  10. Courcelle, B., Makowsky, J.A., Rotics, U.: On the fixed parameter complexity of graph enumeration problems definable in monadic second order logic. Discrete Applied Mathematics 108(1–2), 23–52 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  11. Courcelle, B., Olariu, S.: Upper bounds to the clique–width of graphs. Discrete Applied Mathematics 101, 77–114 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Dong, F.M., et al.: The vertex-cover polynomial of a graph. Discrete Mathematics 250, 71–78 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Dong, F.M., Koh, K.M., Teo, K.L.: Chromatic Polynomials and Chromaticity of Graphs. World Scientific, Singapore (2005)

    MATH  Google Scholar 

  14. Ebbinghaus, H.D., Flum, J.: Finite Model Theory. In: Perspectives in Mathematical Logic, Springer, Heidelberg (1995)

    Google Scholar 

  15. Fischer, E., Makowsky, J.A.: On spectra of sentences of monadic second order logic with counting. Journal of Symbolic Logic 69(3), 617–640 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  16. Frick, M., Grohe, M.: The complexity of first-order and monadic second-order logic revisited. Annals of Pure and Applied Logic 130(1), 3–31 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Golumbic, M.C., Rotics, U.: On the clique-width of some perfect graph classes. Internation Journal of Foundations of Computer Science 11, 423–443 (2000)

    Article  MathSciNet  Google Scholar 

  18. Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)

    MATH  Google Scholar 

  19. Gurevich, Y.: Modest theory of short chains, I. Journal of Symbolic Logic 44, 481–490 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  20. Gurski, F., Wanke, E.: On the relationship between NLC-width and linear NLC-width. Theoretical Computer Science 347(1–2), 76–89 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kobrinski, N.E., Trakhtenbrot, B.A.: Introduction to the Theory of Finite Automata. In: Studies in Logic and the Foundations of Mathematics, North-Holland, Amsterdam (1965)

    Google Scholar 

  22. Läuchli, H.: A decision procedure for the weak second order theory of linear order. In: Logic Colloquium 1966, pp. 189–197. North-Holland, Amsterdam (1968)

    Google Scholar 

  23. Makowsky, J.A.: Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic 126, 1–3 (2004)

    Article  MathSciNet  Google Scholar 

  24. Makowsky, J.A.: From a zoo to a zoology: Towards a general theory of graph polynomials. Theory of Computing Systems, published online first (July 2007)

    Google Scholar 

  25. Makowsky, J.A., Mariño, J.P.: Farrell polynomials on graphs of bounded treewidth. Advances in Applied Mathematics 30, 160–176 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  26. Makowsky, J.A., Zilber, B.: Polynomial invariants of graphs and totally categorical theories. MODNET Preprint No. 21 server (2006), http://www.logique.jussieu.fr/modnet/Publications/Preprint%20

  27. Noy, M., Ribó, A.: Recursively constructible families of graphs. Advances in Applied Mathematics 32, 350–363 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  28. Rozenberg, G. (ed.): Handbook of graph grammars and computing by graph transformations, Foundations, vol. 1. World Scientific, Singapore (1997)

    Google Scholar 

  29. Shelah, S.: The monadic theory of order. Annals of Mathematics 102, 379–419 (1975)

    Article  MathSciNet  Google Scholar 

  30. Trakhtenbrot, B.: Finite automata and the logic of monadic predicates. Doklady Akademy Nauk SSSR 140, 326–329 (1961)

    Google Scholar 

  31. Trakhtenbrot, B.: Some constructions in the monadic predicate calculus. Doklady Akademy Nauk SSSR 138, 320–321 (1961)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Computer Science, Technion–Israel Institute of Technology, Haifa, Israel

    Eldar Fischer & Johann A. Makowsky

Authors
  1. Eldar Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Johann A. Makowsky
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Arnon Avron Nachum Dershowitz Alexander Rabinovich

Rights and permissions

Reprints and permissions

Copyright information

© 2008 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Fischer, E., Makowsky, J.A. (2008). Linear Recurrence Relations for Graph Polynomials. In: Avron, A., Dershowitz, N., Rabinovich, A. (eds) Pillars of Computer Science. Lecture Notes in Computer Science, vol 4800. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-78127-1_15

Download citation

  • DOI: https://doi.org/10.1007/978-3-540-78127-1_15

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-78126-4

  • Online ISBN: 978-3-540-78127-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation