Algorithmica

, Volume 78, Issue 1, pp 342–377 | Cite as

Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm

  • Isolde Adler
  • Mamadou Moustapha Kanté
  • O-joung Kwon
Article

Abstract

Linear rank-width is a linearized variation of rank-width, and it is deeply related to matroid path-width. In this paper, we show that the linear rank-width of every n-vertex distance-hereditary graph, equivalently a graph of rank-width at most 1, can be computed in time \({\mathcal {O}}(n^2\cdot \log _2 n)\), and a linear layout witnessing the linear rank-width can be computed with the same time complexity. As a corollary, we show that the path-width of every n-element matroid of branch-width at most 2 can be computed in time \({\mathcal {O}}(n^2\cdot \log _2 n)\), provided that the matroid is given by its binary representation. To establish this result, we present a characterization of the linear rank-width of distance-hereditary graphs in terms of their canonical split decompositions. This characterization is similar to the known characterization of the path-width of forests given by Ellis, Sudborough, and Turner [The vertex separation and search number of a graph. Inf. Comput., 113(1):50–79, 1994]. However, different from forests, it is non-trivial to relate substructures of the canonical split decomposition of a graph with some substructures of the given graph. We introduce a notion of ‘limbs’ of canonical split decompositions, which correspond to certain vertex-minors of the original graph, for the right characterization.

Keywords

Rank-width Linear rank-width Distance-hereditary graphs Vertex-minors Matroid branch-width Matroid path-width 

References

  1. 1.
    Adler, I., Kanté, Mamadou Moustapha, O-joung, K.: Linear rank-width of distance-hereditary graphs II. Vertex-minor obstructions. Submitted, (2015). http://arxiv.org/abs/1508.04718
  2. 2.
    Bandelt, H.J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory Ser. B 41(2), 182–208 (1986)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, and minimum elimination tree height. J. Algorithms 18, 238–255 (1991)CrossRefMATHGoogle Scholar
  4. 4.
    Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bouchet, A.: Transforming trees by successive local complementations. J. Graph Theory 12(2), 195–207 (1988)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–114 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Courcelle, Bo, Oum, S.: Vertex-minors, monadic second-order logic, and a conjecture by Seese. J. Comb. Theory, Ser. B 97(1), 91–126 (2007)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Cunnigham, W.H., Edmonds, J.: A combinatorial decomposition theory. Can. J. Math. 32, 734–765 (1980)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    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
  10. 10.
    Dharmatilake, J.S.: A min-max theorem using matroid separations. In: Matroid theory (Seattle, WA, 1995), volume 197 of Contemp. Math., pp. 333–342. Amer. Math. Soc., Providence, RI, (1996)Google Scholar
  11. 11.
    Diestel, R.: Graph theory, volume 173 of Graduate Texts in Mathematics. Springer, Heidelberg, fourth edition, (2010)Google Scholar
  12. 12.
    Ganian, R.: Thread graphs, linear rank-width and their algorithmic applications. In: Iliopoulos, Costas S., Smyth, William F. (eds.), IWOCA, volume 6460 of Lecture Notes in Computer Science, pp. 38–42. Springer, (2010)Google Scholar
  13. 13.
    Gavoille, C., Paul, C.: Distance labeling scheme and split decomposition. Discrete Math. 273(1–3), 115–130 (2003)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Geelen, J.F., Gerards, A.M.H., Whittle, G.: Branch-width and well-quasi-ordering in matroids and graphs. J. Combin. Theory Ser. B 84(2), 270–290 (2002)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Geelen, J., Gerards, B., Whittle, G.: Excluding a planar graph from \(\text{ GF }(q)\) representable matroids. J. Combin. Theory Ser. B 97(6), 971–998 (2007)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Geelen, J., Gerards, B., Whittle, G.: Solving Rota’s conjecture. Notices Amer. Math. Soc. 61(7), 736–743 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Geelen, J.F., Oum, S.: Circle graph obstructions under pivoting. J. Graph Theory 61(1), 1–11 (2009)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Isolde Adler and Mamadou Moustapha Kanté: Linear rank-width and linear clique-width of trees. Theoret. Comput. Sci. 589, 87–98 (2015)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Jeong, J., Kim, E.J., Oum, S.: Constructive algorithm for path-width of matroids. SODA: 1695–1704 (2016)Google Scholar
  20. 20.
    Jeong, J., Kwon, O., Oum, S.: Excluded vertex-minors for graphs of linear rank-width at most \(k\). European J. Combin. 41, 242–257 (2014)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Jonathan, A., 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
  22. 22.
    Kashyap, N.: Matroid pathwidth and code trellis complexity. SIAM J. Discrete Math. 22(1), 256–272 (2008)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Kloks,T., Bodlaender, H.L., Müller, H., Kratsch, D.: Computing treewidth and minimum fill-in: All you need are the minimal separators. In: Lengauer, Thomas (ed.), ESA, volume 726 of Lecture Notes in Computer Science, pp. 260–271. Springer, (1993)Google Scholar
  24. 24.
    Mamadou Moustapha Kanté and Michael Rao: The rank-width of edge-coloured graphs. Theory Comput. Syst. 52(4), 599–644 (2013)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Mamadou Moustapha Kanté: Well-quasi-ordering of matrices under Schur complement and applications to directed graphs. European J. Combin. 33(8), 1820–1841 (2012)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    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)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Oum, S.: Rank-width and vertex-minors. J. Comb. Theory, Ser. B 95(1), 79–100 (2005)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Oum, S., Seymour, P.: Approximating clique-width and branch-width. J. Comb. Theory, Ser. B 96(4), 514–528 (2006)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Oxley, J.: Matroid Theory. Number 21 in Oxford Graduate Texts in Mathematics. Oxford University Press, second edition, (2012)Google Scholar
  30. 30.
    Tutte, W.T.: A homotopy theorem for matroids. I, II. Trans. Amer. Math. Soc. 88, 144–174 (1958)MathSciNetMATHGoogle Scholar
  31. 31.
    Tutte, W.T.: Lectures on matroids. J. Res. Nat. Bur. Standards Sect. B 69B, 1–47 (1965)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Isolde Adler
    • 1
  • Mamadou Moustapha Kanté
    • 2
  • O-joung Kwon
    • 3
    • 4
  1. 1.School of ComputingUniversity of LeedsLeedsUK
  2. 2.Université Clermont Auvergne, Université Blaise Pascal, LIMOS, CNRSAubièreFrance
  3. 3.Department of Mathematical SciencesKAISTYuseong-gu, DaejeonSouth Korea
  4. 4.Institute for Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI)BudapestHungary

Personalised recommendations