Linear Rank-Width of Distance-Hereditary Graphs

  • Isolde Adler
  • Mamadou Moustapha Kanté
  • O-joung Kwon
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8747)

Abstract

We present a characterization of the linear rank-width of distance-hereditary graphs. Using the characterization, we show that the linear rank-width of every \(n\)-vertex distance-hereditary graph can be computed in time \(\mathcal {O}(n^2\cdot \log (n))\), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. For our characterization, we combine modifications of canonical split decompositions with an idea of [Megiddo, Hakimi, Garey, Johnson, Papadimitriou: The complexity of searching a graph. JACM 1988], used for computing the path-width of trees. We also provide a set of distance-hereditary graphs which contains the set of distance-hereditary vertex-minor obstructions for linear rank-width. The set given in [Jeong, Kwon, Oum: Excluded vertex-minors for graphs of linear rank-width at most k. STACS 2013: 221–232] is a subset of our obstruction set.

References

  1. 1.
    Adler, I., Farley, A.M., Proskurowski, A.: Obstructions for linear rank-width at most 1. Discrete Appl. Math. 168, 3–13 (2014)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Adler, I., Kanté, M.M.: Linear rank-width and linear clique-width of trees. In: Brandstädt, A., Jansen, K., Reischuk, R. (eds.) WG 2013. LNCS, vol. 8165, pp. 12–25. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  3. 3.
    Bandelt, H.-J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory, Ser. B 41(2), 182–208 (1986)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bouchet, A.: Transforming trees by successive local complementations. J. Graph Theory 12(2), 195–207 (1988)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–114 (2000)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Cunnigham, W.H., Edmonds, J.: A combinatorial decomposition theory. Can. J. Math. 32, 734–765 (1980)CrossRefGoogle Scholar
  7. 7.
    Dahlhaus, E.: Parallel algorithms for hierarchical clustering, and applications to split decomposition and parity graph recognition. J. Graph Algorithms 36(2), 205–240 (2000)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Diestel, R.: Graph Theory. Graduate texts in mathematics, vol. 173, 3rd edn. Springer, Heidelberg (2005)MATHGoogle Scholar
  9. 9.
    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)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fellows, M.R., Rosamond, F.A., Rotics, U., Szeider, S.: Clique-width is np-complete. SIAM J. Discrete Math. 23(2), 909–939 (2009)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Gavoille, C., Paul, C.: Distance labeling scheme and split decomposition. Discrete Math. 273(1–3), 115–130 (2003)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Gioan, E., Paul, C.: Split decomposition and graph-labelled trees: characterizations and fully dynamic algorithms for totally decomposable graphs. Discrete Appl. Math. 160(6), 708–733 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Kloks, T., Bodlaender, H.L., Müller, H., Kratsch, D.: Computing treewidth and minimum fill-in: All you need are the minimal separators. In: Lengauer, T. (ed.) ESA 1993. LNCS, vol. 726, pp. 260–271. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  16. 16.
    Megiddo, N., Louis Hakimi, S., Garey, M.R., Johnson, D.S., Papadimitriou, C.H.: The complexity of searching a graph. J. ACM 35(1), 18–44 (1988)CrossRefMATHGoogle Scholar
  17. 17.
    Oum, S.: Rank-width and vertex-minors. J. Comb. Theory, Ser. B 95(1), 79–100 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Oum, S., Seymour, P.D.: Approximating clique-width and branch-width. J. Comb. Theory, Ser. B 96(4), 514–528 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Isolde Adler
    • 1
  • Mamadou Moustapha Kanté
    • 2
  • O-joung Kwon
    • 3
  1. 1.Institut für InformatikGoethe-UniversitätFrankfurtGermany
  2. 2.Clermont-Université, Université Blaise Pascal, LIMOS, CNRSClermont-FerrandFrance
  3. 3.Department of Mathematical SciencesKAISTDaejeonSouth Korea

Personalised recommendations