Abstract
Connection matrices of graph parameters were first introduced by M. Freedman, L. Lovász and A. Schrijver (2007) to study the question which graph parameters can be represented as counting functions of weighted homomorphisms. The rows and columns of a connection matrix \(M(f,\Box)\) of a graph parameter f and a binary operation \(\Box\) are indexed by all finite (labeled) graphs Gi and the entry at (Gi,Gj) is given by the value of \(f(Gi \Box Gj )\). Connection matrices turned out to be a very powerful tool for studying graph parameters in general.
B. Godlin, T. Kotek and J.A. Makowsky (2008) noticed that connection matrices can be defined for general relational structures and binary operations between them, and for general structural parameters. They proved that for structural parameters f definable in Monadic Second Order Logic, (MSOL) and binary operations compatible with MSOL, the connection matrix \(M(f,\Box)\) has always finite rank. In this talk we discuss several applications of this Finite Rank Theorem, and outline ideas for further research.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial of a graph. Journal of Combinatorial Theory, Series B 92, 199–233 (2004)
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)
Courcelle, B., Olariu, S.: Upper bounds to the clique–width of graphs. Discrete Applied Mathematics 101, 77–114 (2000)
Courcelle, B.: A multivariate interlace polynomial (preprint) (December 2006)
Diestel, R.: Graph Theory. Graduate Texts in Mathematics. Springer, Heidelberg (1996)
de Rougemont, M.: Uniform definability on finite structures with successor. In: SIGACT 1984, pp. 409–417 (1984)
Edwards, K.: The harmonious chromatic number and the achromatic number. In: Bailey, R.A. (ed.) Survey in Combinatorics. London Math. Soc. Lecture Note Ser., vol. 241, pp. 13–47. Cambridge Univ. Press, Cambridge (1997)
Edwards, K., McDiarmid, C.: The complexity of harmonious colouring for trees. Discrete Appl. Math. 57(2-3), 133–144 (1995)
Freedman, M., Lovász, L., Schrijver, A.: Reflection positivity, rank connectivity, and homomorphisms of graphs. Journal of AMS 20, 37–51 (2007)
Godlin, B., Kotek, T., Makowsky, J.A.: Evaluation of graph polynomials. In: International Symposium on Theoretical Programming. LNCS. Springer, Heidelberg (2008)
Goodall, A.J.: Some new evaluations of the Tutte polynomial. Journal of Combinatorial Theory, Series B 96, 207–224 (2006)
Goodall, A.J.: Parity, eulerian subgraphs and the Tutte polynomial. Journal of Combinatorial Theory, Series B 98(3), 599–628 (2008)
Godsil, C., Royle, G.: Algebraic Graph Theory. Graduate Texts in Mathematics. Springer, Heidelberg (2001)
Harary, F., Hedetniemi, S., Prins, G.: An interpolation theorem for graphical homomorphisms. Portugal. Math. 26, 453–462 (1967)
Hopcroft, J.E., Krishnamoorthy, M.S.: On the harmonious coloring of graphs. SIAM J. Algebraic Discrete Methods 4, 306–311 (1983)
Hughes, F., MacGillivray, G.: The achromatic number of graphs: a survey and some new results. Bull. Inst. Combin. Appl. 19, 27–56 (1997)
Kotek, T., Makowsky, J.A., Zilber, B.: On counting generalized colorings. In: Kaminski, M., Martini, S. (eds.) CSL 2008. LNCS, vol. 5213, pp. 339–353. Springer, Heidelberg (2008)
Linial, M.: Hard enumeration problems in geometry and combinatorics. SIAM Journal of Algebraic and Discrete Methods 7, 331–335 (1986)
Linial, N., Matousek, J., Sheffet, O., Tardos, G.: Graph coloring with no large monochromatic components. arXiv:math/0703362 (2007)
Lovász, L.: Connection matrics. In: Grimmet, G., McDiarmid, C. (eds.) Combinatorics, Complexity and Chance, A Tribute to Dominic Welsh, pp. 179–190. Oxford University Press, Oxford (2007)
Makowsky, J.A.: Algorithmic uses of the Feferman-Vaught theorem. Annals of Pure and Applied Logic 126(1-3), 159–213 (2004)
Makowsky, J.A.: Colored Tutte polynomials and Kauffman brackets on graphs of bounded tree width. Disc. Appl. Math. 145(2), 276–290 (2005)
Makowsky, J.A.: From a zoo to a zoology: Towards a general theory of graph polynomials. Theory of Computing Systems (July 2007), http://www.dx.doi.org/10.1017/s00224-007-9022-9
Moran, S., Snir, S.: Efficient approximation of convex recolorings. Journal of Computer and System Sciences 73(7), 1078–1089 (2007)
Makowsky, J.A., Zilber, B.: Polynomial invariants of graphs and totally categorical theories. MODNET Preprint No. 21 (2006), http://www.logique.jussieu.fr/modnet/Publications/Preprint%20server
Oum, S.: Approximating rank-width and clique-width quickly. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 49–58. Springer, Heidelberg (2005)
Schrijver, A.: Polynomial and tensor invariants and combinatorial parameters. Amsterdam (preprint, 2008)
Szegedy, B.: Edge coloring models and reflection positivity, arXiv: math.CO/0505035 (2007)
Welsh, D.J.A.: Complexity: Knots, Colourings and Counting. London Mathematical Society Lecture Notes Series, vol. 186. Cambridge University Press, Cambridge (1993)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2008 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Makowsky, J. (2008). Connection Matrices for MSOL-Definable Structural Invariants. In: Ramanujam, R., Sarukkai, S. (eds) Logic and Its Applications. ICLA 2009. Lecture Notes in Computer Science(), vol 5378. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-92701-3_4
Download citation
DOI: https://doi.org/10.1007/978-3-540-92701-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-92700-6
Online ISBN: 978-3-540-92701-3
eBook Packages: Computer ScienceComputer Science (R0)