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.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aigner, M., van der Holst, H.: Interlace polynomials. Linear Algebra and its Applications 377, 11–30 (2004)
Arratia, R., Bollobás, B., Sorkin, G.B.: The interlace polynomial of a graph. Journal of Combinatorial Theory, Series B 92(2), 199–233 (2004)
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)
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)
Brijder, R., Hoogeboom, H.J.: Interlace polynomials for multimatroids and delta-matroids. European Journal of Combinatorics 40, 142–167 (2014)
Burns, J., Dolzhenko, E.: Assembly words (properties), http://knot.math.usf.edu/assembly/properties.html (visited March 2014)
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)
Cohn, M., Lempel, A.: Cycle decomposition by disjoint transpositions. Journal of Combinatorial Theory, Series A 13(1), 83–89 (1972)
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)
Ehrenfeucht, A., Harju, T., Petre, I., Prescott, D.M., Rozenberg, G.: Computation in Living Cells – Gene Assembly in Ciliates. Springer (2004)
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)
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)
Ellis-Monaghan, J.A., Sarmiento, I.: Generalized transition polynomials. Congressus Numerantium 155, 57–69 (2002)
Godsil, C., Royle, G.: Algebraic Graph Theory. Springer (2001)
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)
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)
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/
Prescott, D.M.: Genome gymnastics: Unique modes of DNA evolution and processing in ciliates. Nature Reviews 1, 191–199 (2000)
Prescott, D.M., Greslin, A.F.: Scrambled actin I gene in the micronucleus of Oxytricha nova. Developmental Genetics 13, 66–74 (1992)
Traldi, L.: Binary nullity, Euler circuits and interlace polynomials. European Journal of Combinatorics 32(6), 944–950 (2011)
Traldi, L., Zulli, L.: A bracket polynomial for graphs. I. Journal of Knot Theory and Its Ramifications 18(12), 1681–1709 (2009)
Tsatsomeros, M.J.: Principal pivot transforms: properties and applications. Linear Algebra and its Applications 307(1-3), 151–165 (2000)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Brijder, R., Hoogeboom, H.J. (2014). Graph Polynomials Motivated by Gene Rearrangements in Ciliates. In: Beckmann, A., Csuhaj-Varjú, E., Meer, K. (eds) Language, Life, Limits. CiE 2014. Lecture Notes in Computer Science, vol 8493. Springer, Cham. https://doi.org/10.1007/978-3-319-08019-2_7
Download citation
DOI: https://doi.org/10.1007/978-3-319-08019-2_7
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08018-5
Online ISBN: 978-3-319-08019-2
eBook Packages: Computer ScienceComputer Science (R0)