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Linear Rank-Width and Linear Clique-Width of Trees

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Book cover Graph-Theoretic Concepts in Computer Science (WG 2013)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 8165))

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Abstract

We show that for every forest T the linear rank-width of T is equal to the path-width of T, and we show that the linear clique-width of T equals the path-width of T plus two, provided that T contains a path of length three. It follows that both linear rank-width and linear clique-width of forests can be computed in linear time. Using our characterization of linear rank-width of forests, we determine the set of minimal excluded acyclic vertex-minors for the class of graphs of linear rank-width at most k.

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References

  1. Adler, I., Farley, A.M., Proskurowski, A.: Obstructions for linear rankwidth at most 1. J. Discrete Applied Mathematics (to appear, 2013)

    Google Scholar 

  2. Bienstock, D., Seymour, P.D.: Monotonicity in graph searching. J. Algorithms 12(2), 239–245 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bodlaender, H.L., Kloks, T., Kratsch, D.: Treewidth and pathwidth of permutation graphs. SIAM J. Discrete Math. 8(4), 606–616 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bodlaender, H.L., Möhring, R.H.: The pathwidth and treewidth of cographs. In: Gilbert, J.R., Karlsson, R. (eds.) SWAT 1990. LNCS, vol. 447, pp. 301–309. Springer, Heidelberg (1990)

    Google Scholar 

  6. Bouchet, A.: Transforming trees by successive local complementations. J. Graph Theory 12(2), 195–207 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bouchet, A.: Circle graph obstructions. J. Comb. Theory, Ser. B 60(1), 107–144 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  8. Courcelle, B., Engelfriet, J.: Graph Structure and Monadic Second-Order Logic, A Language-Theoretic Approach. Cambridge University Press (2012)

    Google Scholar 

  9. Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Applied Mathematics 101(1-3), 77–114 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ellis, J.A., Sudborough, I.H., Turner, J.S.: The vertex separation and search number of a graph. Inf. Comput. 113(1), 50–79 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Fellows, M.R., Rosamond, F.A., Rotics, U., Szeider, S.: Clique-width is np-complete. SIAM J. Discrete Math. 23(2), 909–939 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ganian, R.: Thread graphs, linear rank-width and their algorithmic applications. In: Iliopoulos, C.S., Smyth, W.F. (eds.) IWOCA 2010. LNCS, vol. 6460, pp. 38–42. Springer, Heidelberg (2011)

    Chapter  Google Scholar 

  13. Gurski, F.: Characterizations for co-graphs defined by restricted nlc-width or clique-width operations. Discrete Mathematics 306(2), 271–277 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gurski, F.: Linear layouts measuring neighbourhoods in graphs. Discrete Mathematics 306(15), 1637–1650 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Gurski, F., Wanke, E.: On the relationship between nlc-width and linear nlc-width. Theor. Comput. Sci. 347(1-2), 76–89 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gurski, F., Wanke, E.: The nlc-width and clique-width for powers of graphs of bounded tree-width. Discrete Applied Mathematics 157(4), 583–595 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Heggernes, P., Meister, D., Papadopoulos, C.: A complete characterisation of the linear clique-width of path powers. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 241–250. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  18. Heggernes, P., Meister, D., Papadopoulos, C.: Graphs of linear clique-width at most 3. Theor. Comput. Sci. 412(39), 5466–5486 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. Heggernes, P., Meister, D., Papadopoulos, C.: Characterising the linear clique-width of a class of graphs by forbidden induced subgraphs. Discrete Applied Mathematics 160(6), 888–901 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  20. Oum, S.I.: Rank-width and vertex-minors. J. Comb. Theory, Ser. B 95(1), 79–100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Oum, S.: Rank-width and well-quasi-ordering. SIAM J. Discrete Math. 22(2), 666–682 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jeong, J., Kwon, O.-J., Oum, S.I.: Excluded vertex-minors for graphs of linear rank-width at most k. In: Portier, N., Wilke, T. (eds.) STACS. LIPIcs, vol. 20, pp. 221–232. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2013)

    Google Scholar 

  23. Kirousis, L.M., Papadimitriou, C.H.: Searching and pebbling. Theor. Comput. Sci. 47(3), 205–218 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kloks, T., Bodlaender, H.L.: Approximating treewidth and pathwidth of some classes of perfect graphs. In: Ibaraki, T., Inagaki, Y., Iwama, K., Nishizeki, T., Yamashita, M. (eds.) ISAAC 1992. LNCS, vol. 650, pp. 116–125. Springer, Heidelberg (1992)

    Chapter  Google Scholar 

  25. Lagergren, J.: Upper bounds on the size of obstructions and intertwines. J. Comb. Theory, Ser. B 73(1), 7–40 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. LaPaugh, A.S.: Recontamination does not help to search a graph. J. ACM 40(2), 224–245 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Lozin, V.V., Rautenbach, D.: The relative clique-width of a graph. J. Comb. Theory, Ser. B 97(5), 846–858 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  28. Megiddo, N., Hakimi, S.L., Garey, M.R., Johnson, D.S., Papadimitriou, C.H.: The complexity of searching a graph. J. ACM 35(1), 18–44 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  29. Oum, S.-I., Seymour, P.D.: Approximating clique-width and branch-width. J. Comb. Theory, Ser. B 96(4), 514–528 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Scheffler, P.: Die Baumweite von Graphen als ein Maß für die Kompliziertheit algorithmischer Probleme. Akademie der Wissenschaften der DDR, Berlin, PhD thesis (1989)

    Google Scholar 

  31. Suchan, K., Todinca, I.: Pathwidth of circular-arc graphs. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 258–269. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  32. Takahashi, A., Ueno, S., Kajitani, Y.: Minimal acyclic forbidden minors for the family of graphs with bounded path-width. Discrete Mathematics 127(1-3), 293–304 (1994)

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Informatik, Goethe-Universität, Frankfurt, Germany

    Isolde Adler

  2. LIMOS, CNRS, Clermont-Université, Université Blaise Pascal, France

    Mamadou Moustapha Kanté

Authors
  1. Isolde Adler
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  2. Mamadou Moustapha Kanté
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Editor information

Editors and Affiliations

  1. Institute of Computer Science, University of Rostock, Albert-Einstein-Straße 22, 18059, Rostock, Germany

    Andreas Brandstädt

  2. Department of Computer Science, University of Kiel, Olshausenstraße 40, 24098, Kiel, Germany

    Klaus Jansen

  3. Institute of Theoretical Computer Science, University of Lübeck, Ratzeburger Allee 160, 23538, Lübeck, Germany

    Rüdiger Reischuk

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Adler, I., Kanté, M.M. (2013). Linear Rank-Width and Linear Clique-Width of Trees. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds) Graph-Theoretic Concepts in Computer Science. WG 2013. Lecture Notes in Computer Science, vol 8165. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45043-3_3

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  • DOI: https://doi.org/10.1007/978-3-642-45043-3_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-45042-6

  • Online ISBN: 978-3-642-45043-3

  • eBook Packages: Computer ScienceComputer Science (R0)

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