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On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions

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Computing and Combinatorics (COCOON 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9198))

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Abstract

Symbolic ultrametrics define edge-colored complete graphs \(K_n\) and yield a simple tree representation of \(K_n\). We discuss, under which conditions this idea can be generalized to find a symbolic ultrametric that, in addition, distinguishes between edges and non-edges of arbitrary graphs \(G=(V,E)\) and thus, yielding a simple tree representation of G. We prove that such a symbolic ultrametric can only be defined for G if and only if G is a so-called cograph. A cograph is uniquely determined by a so-called cotree. As not all graphs are cographs, we ask, furthermore, what is the minimum number of cotrees needed to represent the topology of G. The latter problem is equivalent to find an optimal cograph edge k-decomposition \(\{E_1,\dots ,E_k\}\) of E so that each subgraph \((V,E_i)\) of G is a cograph. An upper bound for the integer k is derived and it is shown that determining whether a graph has a cograph 2-decomposition, resp., 2-partition is NP-complete.

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References

  1. Achlioptas, D.: The complexity of g-free colourability. Discrete Mathematics 165–166, 21–30 (1997). Graphs and Combinatorics

    Article  MathSciNet  Google Scholar 

  2. Bernini, A., Ferrari, L., Pinzani, R.: Enumeration of some classes of words avoiding two generalized patterns of length three. arXiv preprint arXiv:0711.3387 (2007)

  3. Bilotta, S., Grazzini, E., Pergola, E., Morgagni, V.G.B.: Counting binary words avoiding alternating patterns. Journal of Integer Sequences 16(2), 3 (2013)

    Google Scholar 

  4. Böcker, S., Dress, A.W.M.: Recovering symbolically dated, rooted trees from symbolic ultrametrics. Adv. Math. 138, 105–125 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Brändén, P., Mansour, T.: Finite automata and pattern avoidance in words. Journal of Combinatorial Theory, Series A 110(1), 127–145 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. Brandstädt, A., Le, V.B., Spinrad, J.P.: Graph Classes: A Survey. SIAM Monographs on Discrete Mathematics and Applications. Soc. Ind. Appl. Math., Philadephia (1999)

    Google Scholar 

  7. Burstein, A., Mansour, T.: Words restricted by patterns with at most 2 distinct letters. Electron. J. Combin. Number Theory 9(2), 1–16 (2002)

    Google Scholar 

  8. Corneil, D.G., Lerchs, H., Burlingham, L.K.S.: Complement reducible graphs. Discr. Appl. Math. 3, 163–174 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  9. Corneil, D.G., Perl, Y., Stewart, L.K.: A linear recognition algorithm for cographs. SIAM Journal on Computing 14(4), 926–934 (1985)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dorbec, P., Montassier, M., Ochem, P.: Vertex partitions of graphs into cographs and stars. Journal of Graph Theory 75(1), 75–90 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  11. Gimbel, J., Nesětrǐl, J.: Partitions of graphs into cographs. Electronic Notes in Discrete Mathematics 11, 705–721 (2002). The Ninth Quadrennial International Conference on Graph Theory, Combinatorics, Algorithms and Applications

    Article  Google Scholar 

  12. Hammack, R., Imrich, W., Klavžar, S.: Handbook of Product graphs, 2nd edn. CRC Press, Boca Raton (2011)

    MATH  Google Scholar 

  13. Hellmuth, M., Hernandez-Rosales, M., Huber, K.T., Moulton, V., Stadler, P.F., Wieseke, N.: Orthology relations, symbolic ultrametrics, and cographs. Journal of Mathematical Biology 66(1–2), 399–420 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Hellmuth, M., Wiesecke, N., Lenhof, H.P., Middendorf, M., Stadler, P.F.: Phylogenomics with paralogs. PNAS 112(7), 2058–2063 (2015)

    Article  Google Scholar 

  15. Lafond, M., El-Mabrouk, N.: Orthology and paralogy constraints: satisfiability and consistency. BMC Genomics 15(Suppl. 6), S12 (2014)

    Article  Google Scholar 

  16. Lerchs, H.: On cliques and kernels. Technical report, Dept. of Comput. Sci. University of Toronto (1971)

    Google Scholar 

  17. Lerchs, H.: On the clique-kernel structure of graphs. Technical report, Dept. of Comput. Sci. University of Toront (1971)

    Google Scholar 

  18. Liu, Y., Wang, J., Guo, J., Chen, J.: Cograph editing: complexity and parameterized algorithms. In: Fu, B., Du, D.-Z. (eds.) COCOON 2011. LNCS, vol. 6842, pp. 110–121. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  19. Liu, Y., Wang, J., Guo, J., Chen, J.: Complexity and parameterized algorithms for cograph editing. Theoretical Computer Science 461, 45–54 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Misra, J., Gries, D.: A constructive proof of vizing’s theorem. Information Processing Letters 41(3), 131–133 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  21. Moret, B.M.: The Theory of Computation. Addison-Wesley (1997)

    Google Scholar 

  22. Pudwell, L.K.: Enumeration schemes for pattern-avoiding words and permutations. ProQuest (2008)

    Google Scholar 

  23. Pudwell, L.K.: Enumeration schemes for words avoiding patterns with repeated letters. Electron. J. Combin. Number Theory 8(A40), 1–19 (2008)

    MathSciNet  Google Scholar 

  24. Schaefer, T.J.: The complexity of satisfiability problems. In: Proceedings of the Tenth Annual ACM Symposium on Theory of Computing, STOC 1978, pp. 216–226. ACM, New York (1978)

    Google Scholar 

  25. Vizing, V.G.: On an estimate of the chromatic class of a p-graph. Journal of Mathematical Biology 3, 23–30 (1964). (Russian)

    MathSciNet  Google Scholar 

  26. Zhang, P.: A study on generalized solution concepts in constraint satisfaction and graph colouring. Master’s thesis, University of British Columbia, Canada (2014)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Greifswald, Walther-Rathenau-Strasse 47, 17487, Greifswald, Germany

    Marc Hellmuth

  2. Center for Bioinformatics, Saarland University, Building E 2.1, 66041, Saarbrücken, Germany

    Marc Hellmuth

  3. Parallel Computing and Complex Systems Group, Department of Computer Science, Leipzig University, Augustusplatz 10, 04109, Leipzig, Germany

    Nicolas Wieseke

Authors
  1. Marc Hellmuth
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  2. Nicolas Wieseke
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Correspondence to Marc Hellmuth .

Editor information

Editors and Affiliations

  1. Beijing University of Technology, Beijing, China

    Dachuan Xu

  2. University of New Brunswick, Fredericton, Canada

    Donglei Du

  3. University of Texas at Dallas, Richardson, USA

    Dingzhu Du

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Hellmuth, M., Wieseke, N. (2015). On Symbolic Ultrametrics, Cotree Representations, and Cograph Edge Decompositions and Partitions. In: Xu, D., Du, D., Du, D. (eds) Computing and Combinatorics. COCOON 2015. Lecture Notes in Computer Science(), vol 9198. Springer, Cham. https://doi.org/10.1007/978-3-319-21398-9_48

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  • DOI: https://doi.org/10.1007/978-3-319-21398-9_48

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21397-2

  • Online ISBN: 978-3-319-21398-9

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