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

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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.

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Notes

  1. At the time this paper was submitted, the algorithm in [19] was not even known.

References

  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. Bandelt, H.J., Mulder, H.M.: Distance-hereditary graphs. J. Comb. Theory Ser. B 41(2), 182–208 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MATH  Google Scholar 

  4. 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 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Courcelle, B., Olariu, S.: Upper bounds to the clique width of graphs. Discrete Appl. Math. 101(1–3), 77–114 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  8. Cunnigham, W.H., Edmonds, J.: A combinatorial decomposition theory. Can. J. Math. 32, 734–765 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  11. Diestel, R.: Graph theory, volume 173 of Graduate Texts in Mathematics. Springer, Heidelberg, fourth edition, (2010)

  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)

  13. Gavoille, C., Paul, C.: Distance labeling scheme and split decomposition. Discrete Math. 273(1–3), 115–130 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  16. Geelen, J., Gerards, B., Whittle, G.: Solving Rota’s conjecture. Notices Amer. Math. Soc. 61(7), 736–743 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  17. Geelen, J.F., Oum, S.: Circle graph obstructions under pivoting. J. Graph Theory 61(1), 1–11 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Isolde Adler and Mamadou Moustapha Kanté: Linear rank-width and linear clique-width of trees. Theoret. Comput. Sci. 589, 87–98 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Jeong, J., Kim, E.J., Oum, S.: Constructive algorithm for path-width of matroids. SODA: 1695–1704 (2016)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kashyap, N.: Matroid pathwidth and code trellis complexity. SIAM J. Discrete Math. 22(1), 256–272 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  24. Mamadou Moustapha Kanté and Michael Rao: The rank-width of edge-coloured graphs. Theory Comput. Syst. 52(4), 599–644 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Oxley, J.: Matroid Theory. Number 21 in Oxford Graduate Texts in Mathematics. Oxford University Press, second edition, (2012)

  30. Tutte, W.T.: A homotopy theorem for matroids. I, II. Trans. Amer. Math. Soc. 88, 144–174 (1958)

    MathSciNet  MATH  Google Scholar 

  31. Tutte, W.T.: Lectures on matroids. J. Res. Nat. Bur. Standards Sect. B 69B, 1–47 (1965)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The authors would like to thank Sang-il Oum for pointing out that the computation of the path-width of matroids of branch-width at most 2 can be obtained as a corollary of our main result.

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Author notes

  1. O-joung Kwon

    Present address: Institute for Computer Science and Control, Hungarian Academy of Sciences (MTA SZTAKI), Kende u.13-17, Budapest, Hungary

Authors and Affiliations

  1. School of Computing, University of Leeds, Leeds, UK

    Isolde Adler

  2. Université Clermont Auvergne, Université Blaise Pascal, LIMOS, CNRS, Aubière, France

    Mamadou Moustapha Kanté

  3. Department of Mathematical Sciences, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon, 305-701, South Korea

    O-joung Kwon

Authors
  1. Isolde Adler
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  2. Mamadou Moustapha Kanté
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  3. O-joung Kwon
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Corresponding author

Correspondence to O-joung Kwon.

Additional information

The first author is supported by the German Research Council, Project GalA, AD 411/1-1. The third author is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2011-0011653). A preliminary version appeared in the proceedings of WG’14.

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Adler, I., Kanté, M.M. & Kwon, Oj. Linear Rank-Width of Distance-Hereditary Graphs I. A Polynomial-Time Algorithm. Algorithmica 78, 342–377 (2017). https://doi.org/10.1007/s00453-016-0164-5

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  • DOI: https://doi.org/10.1007/s00453-016-0164-5

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