Abstract
We report about our ongoing study of inter-reducibilities of parameterized numeric graph invariants and graph polynomials. The purpose of this work is to systematize recent emerging work on graph polynomials and various partition functions with respect to their combinatorial expressiveness and computational complexity.
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References
Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial of a graph. Journal of Combinatorial Theory Series B 92, 199–233 (2004)
Arratia, R., Bollobás, B., Sorkin, G.B.: A two-variable interlace polynomial. Combinatorica 24.4, 567–584 (2004)
Borgs, C., Chayes, J., Lovász, L., Sós, V.T., Vesztergombi, K.: Counting graph homomorphisms. In: Klazar, M., Kratochvil, J., Loebl, M., Matousek, J., Thomas, R., Valtr, P. (eds.) Topics in Discret mathematics, pp. 315–371. Springer, Heidelberg (2006)
Bläser, M., Dell, H.: Complexity of the cover polynomial. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 801–812. Springer, Heidelberg (2007)
Bläser, M., Dell, H., Makowsky, J.A.: Complexity of the Bollobás-Riordan polynomia, exceptional points and uniform reductions. In: CSR 2008 (accepted for presentation) (2008)
Bulatov, A., Grohe, M.: The complexity of partition functions. Theoretical Computer Science 348, 148–186 (2005)
Bläser, M., Hoffmann, C.: On the complexity of the interlace polynomial. arXive 0707.4565 (2007)
Bollobás, B., Riordan, O.: A Tutte polynomial for coloured graphs. Combinatorics, Probability and Computing 8, 45–94 (1999)
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)
Courcelle, B.: A multivariate interlace polynomial (preprint, December 2006)
Dyer, M., Greenhill, C.: The complexity of counting graph homomorphisms. Random Structures and Algorithms 17(3-4), 260–289 (2000)
Freedman, M., Lovász, L., Schrijver, A.: Reflection positivity, rank connectivity, and homomorphisms of graphs. Journal of AMS 20, 37–51 (2007)
Hopcroft, J.E., Krishnamoorthy, M.S.: On the harmonious coloring of graphs. SIAM J. Algebraic Discrete Methods 4, 306–311 (1983)
Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Phil. Soc. 108, 35–53 (1990)
Lotz, M., Makowsky, J.A.: On the algebraic complexity of some families of coloured Tutte polynomials. Advances in Applied Mathematics 32(1-2), 327–349 (2004)
Makowsky, J.A.: From a zoo to a zoology: Towards a general theory of graph polynomials. In: Theory of Computing Systems (on-line first) (2007)
Noble, S.D., Welsh, D.J.A.: A weighted graph polynomial from chromatic invariants of knots. Ann. Inst. Fourier, Grenoble 49, 1057–1087 (1999)
Sokal, A.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Survey in Combinatorics, 2005. London Mathematical Society Lecture Notes, vol. 327, pp. 173–226 (2005)
Szegedy, B.: Edge coloring models and reflection positivity. arXiv: math.CO/0505035 (2007)
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Makowsky, J.A. (2008). Uniform Algebraic Reducibilities between Parameterized Numeric Graph Invariants. In: Beckmann, A., Dimitracopoulos, C., Löwe, B. (eds) Logic and Theory of Algorithms. CiE 2008. Lecture Notes in Computer Science, vol 5028. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-69407-6_43
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DOI: https://doi.org/10.1007/978-3-540-69407-6_43
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