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\(\mathbb F\)-Rank-Width of (Edge-Colored) Graphs

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Algebraic Informatics (CAI 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6742))

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Abstract

Rank-width is a complexity measure equivalent to the clique-width of undirected graphs and has good algorithmic and structural properties. It is in particular related to the vertex-minor relation. We discuss an extension of the notion of rank-width to all types of graphs - directed or not, with edge colors or not -, named \(\mathbb F\)-rank-width. We extend most of the results known for the rank-width of undirected graphs to the \(\mathbb F\)-rank-width of graphs: cubic-time recognition algorithm, characterisation by excluded configurations under vertex-minor and pivot-minor, and algebraic characterisation by graph operations. We also show that the rank-width of undirected graphs is a special case of \(\mathbb F\)-rank-width.

A part of this research is supported by the projects “Graph decompositions and algorithms (GRAAL)” and “Decomposition Of Relational Structures and Combinatorial Optimisation (DORSO)” of French “Agence Nationale Pour la Recherche” and was done when the first author was in Université Bordeaux 1, LaBRI.

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References

  1. Bouchet, A.: Digraph Decompositions and Eulerian Systems. SIAM Journal on Algebraic and Discrete Methods 8(3), 323–337 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  2. Blumensath, A., Courcelle, B.: Recognizability, Hypergraph Operations and Logical Types. Information and Computation 204(6), 853–919 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Courcelle, B.: On the Model-Checking of Monadic Second-Order Formulas with Edge Set Quantifications. Discrete Applied Mathematics doi:10.1016/j.dam.2010.12.017 (in press)

    Google Scholar 

  4. Courcelle, B.: Graph Structure and Monadic Second-Order Logic. Book in preparation. Cambridge University Press, Cambridge

    Google Scholar 

  5. Courcelle, B., Kanté, M.M.: Graph Operations Characterizing Rank-Width. Discrete Applied Mathematics 157(4), 627–640 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Courcelle, B., Makowsky, J.A.: Fusion in Relational Structures and the Verification of Monadic Second-Order Properties. Mathematical Structures in Computer Science 12, 203–235 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. 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 

  8. Courcelle, B., Oum, S.: Vertex-Minors, Monadic Second-Order Logic and a Conjecture by Seese. Journal of Combinatorial Theory, Series B 97(1), 91–126 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Cunningham, W.H.: Decomposition of Directed Graphs. SIAM Journal on Algebraic and Discrete Methods 3(2), 214–228 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  10. Diestel, R.: Graph Theory, 3rd edn. Springer, Heidelberg (2005)

    MATH  Google Scholar 

  11. Ehrenfeucht, A., Harju, T., Rozenberg, G.: The Theory of 2-Structures: A Framework for Decomposition and Transformation of Graphs. World Scientific, Singapore (1999)

    Book  MATH  Google Scholar 

  12. Fisher, E., Makowsky, J.A., Ravve, E.V.: Counting Truth Assignments of Formulas of Bounded Tree-Width or Clique-Width. Discrete Applied Mathematics 156(4), 511–529 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Flarup, U., Lyaudet, L.: On the Expressive Power of Permanents and Perfect Matchings of Matrices of Bounded Path-Width/Clique-Width. Theory of Computing Systems 46(4), 761–791 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  14. Geelen, J.F., Gerards, A.M.H., Robertson, N., Whittle, G.P.: On the Excluded Minors for the Matroids of Branch-Width k. Journal of Combinatorial Theory, Series B 88(2), 261–265 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hliněný, P., Oum, S.: Finding Branch-Decompositions and Rank-Decompositions. SIAM Journal on Computing 38(3), 1012–1032 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kaminski, M., Lozin, V., Milanic, M.: Recent Developments on Graphs of Bounded Clique-Width. Discrete Applied Mathematics 157(12), 2747–2761 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kané, M.M.: Well-Quasi-Ordering of Matrices under Principal Pivot Transforms Revisited, arxiv:1102.2134 (2011) (submitted)

    Google Scholar 

  18. Kanté, M.M., Rao, M.: Directed Rank-Width and Displit Decomposition. In: Habib, M., Paul, C. (eds.) WG 2009. LNCS, vol. 5911, pp. 214–225. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  19. Lidl, R., Niederreiter, H.: Finite Fields. Encyclopedia of Mathematics and its Applications, 2nd edn (1997)

    Google Scholar 

  20. Oum, S.: Rank-Width and Vertex-Minors. Journal of Combinatorial Theory, Series B 95(1), 79–100 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  21. Oum, S., Seymour, P.D.: Approximating Clique-Width and Branch-Width. Journal of Combinatorial Theory, Series B 96(4), 514–528 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Robertson, N., Seymour, P.D.: Graph minors V:Excluding a Planar Graph. Journal of Combinatorial Theory, Series B 41, 92–114 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  23. Schrijver, A.: Combinatorial Optimization, Polyhedra and Efficiency, vol. B. Springer, Heidelberg (2003)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

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

    Mamadou Moustapha Kanté

  2. CNRS - Laboratoire J.V. Poncelet, Moscow, Russia

    Michael Rao

  3. Université de Bordeaux, LaBRI, CNRS, France

    Michael Rao

Authors
  1. Mamadou Moustapha Kanté
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  2. Michael Rao
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Editor information

Editors and Affiliations

  1. Research Institute for Symbolic Computation (RISC), Johannes Kepler University, Altenberger Straße 69, 4040, Linz, Austria

    Franz Winkler

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Kanté, M.M., Rao, M. (2011). \(\mathbb F\)-Rank-Width of (Edge-Colored) Graphs. In: Winkler, F. (eds) Algebraic Informatics. CAI 2011. Lecture Notes in Computer Science, vol 6742. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21493-6_10

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  • DOI: https://doi.org/10.1007/978-3-642-21493-6_10

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-21492-9

  • Online ISBN: 978-3-642-21493-6

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