Graph Polynomials Motivated by Gene Rearrangements in Ciliates

  • Robert Brijder
  • Hendrik Jan Hoogeboom
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8493)

Abstract

Gene rearrangements within the process of gene assembly in ciliates can be represented using a 4-regular graph. Based on this observation, Burns et al. [Discrete Appl. Math., 2013] propose a graph polynomial abstracting basic features of the assembly process, like the number of segments excised. We show that this assembly polynomial is essentially (i) a single variable case of the transition polynomial by Jaeger and (ii) a special case of the bracket polynomial introduced for simple graphs by Traldi and Zulli.

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References

  1. 1.
    Aigner, M., van der Holst, H.: Interlace polynomials. Linear Algebra and its Applications 377, 11–30 (2004)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial of a graph. Journal of Combinatorial Theory, Series B 92(2), 199–233 (2004)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Brijder, R., Daley, M., Harju, T., Jonoska, N., Petre, I., Rozenberg, G.: Computational nature of gene assembly in ciliates. In: Rozenberg, G., Bäck, T., Kok, J. (eds.) Handbook of Natural Computing, vol. 3, pp. 1233–1280. Springer (2012)Google Scholar
  4. 4.
    Brijder, R., Hoogeboom, H.J.: The algebra of gene assembly in ciliates. In: Jonoska, N., Saito, M. (eds.) Discrete and Topological Models in Molecular Biology. Natural Computing Series, pp. 289–307. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  5. 5.
    Brijder, R., Hoogeboom, H.J.: Interlace polynomials for multimatroids and delta-matroids. European Journal of Combinatorics 40, 142–167 (2014)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Burns, J., Dolzhenko, E.: Assembly words (properties), http://knot.math.usf.edu/assembly/properties.html (visited March 2014)
  7. 7.
    Burns, J., Dolzhenko, E., Jonoska, N., Muche, T., Saito, M.: Four-regular graphs with rigid vertices associated to DNA recombination. Discrete Applied Mathematics 161(10-11), 1378–1394 (2013)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Cohn, M., Lempel, A.: Cycle decomposition by disjoint transpositions. Journal of Combinatorial Theory, Series A 13(1), 83–89 (1972)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Dolzhenko, E., Valencia, K.: Invariants of graphs modeling nucleotide rearrangements. In: Jonoska, N., Saito, M. (eds.) Discrete and Topological Models in Molecular Biology. Natural Computing Series, pp. 309–323. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  10. 10.
    Ehrenfeucht, A., Harju, T., Petre, I., Prescott, D.M., Rozenberg, G.: Computation in Living Cells – Gene Assembly in Ciliates. Springer (2004)Google Scholar
  11. 11.
    Ellis-Monaghan, J.A., Merino, C.: Graph polynomials and their applications I: The Tutte polynomial. In: Dehmer, M. (ed.) Structural Analysis of Complex Networks, pp. 219–255. Birkhäuser, Boston (2011)CrossRefGoogle Scholar
  12. 12.
    Ellis-Monaghan, J.A., Merino, C.: Graph polynomials and their applications II: Interrelations and interpretations. In: Dehmer, M. (ed.) Structural Analysis of Complex Networks, pp. 257–292. Birkhäuser, Boston (2011)CrossRefGoogle Scholar
  13. 13.
    Ellis-Monaghan, J.A., Sarmiento, I.: Generalized transition polynomials. Congressus Numerantium 155, 57–69 (2002)MATHMathSciNetGoogle Scholar
  14. 14.
    Godsil, C., Royle, G.: Algebraic Graph Theory. Springer (2001)Google Scholar
  15. 15.
    Jaeger, F.: On transition polynomials of 4-regular graphs. In: Hahn, G., Sabidussi, G., Woodrow, R. (eds.) Cycles and Rays. NATO ASI Series, vol. 301, pp. 123–150. Kluwer (1990)Google Scholar
  16. 16.
    Kotzig, A.: Eulerian lines in finite 4-valent graphs and their transformations. In: Theory of graphs, Proceedings of the Colloquium, Tihany, Hungary, pp. 219–230. Academic Press, New York (1968)Google Scholar
  17. 17.
    Martin, P.: Enumérations eulériennes dans les multigraphes et invariants de Tutte-Grothendieck. PhD thesis, Institut d’Informatique et de Mathématiques Appliquées de Grenoble (IMAG) (1977), http://tel.archives-ouvertes.fr/tel-00287330_v1/
  18. 18.
    Prescott, D.M.: Genome gymnastics: Unique modes of DNA evolution and processing in ciliates. Nature Reviews 1, 191–199 (2000)CrossRefGoogle Scholar
  19. 19.
    Prescott, D.M., Greslin, A.F.: Scrambled actin I gene in the micronucleus of Oxytricha nova. Developmental Genetics 13, 66–74 (1992)CrossRefGoogle Scholar
  20. 20.
    Traldi, L.: Binary nullity, Euler circuits and interlace polynomials. European Journal of Combinatorics 32(6), 944–950 (2011)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Traldi, L., Zulli, L.: A bracket polynomial for graphs. I. Journal of Knot Theory and Its Ramifications 18(12), 1681–1709 (2009)CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Tsatsomeros, M.J.: Principal pivot transforms: properties and applications. Linear Algebra and its Applications 307(1-3), 151–165 (2000)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Robert Brijder
    • 1
  • Hendrik Jan Hoogeboom
    • 2
  1. 1.Hasselt University and Transnational University of LimburgBelgium
  2. 2.Leiden Institute of Advanced Computer ScienceLeiden UniversityThe Netherlands

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