On Digraph Width Measures in Parameterized Algorithmics

  • Robert Ganian
  • Petr Hliněný
  • Joachim Kneis
  • Alexander Langer
  • Jan Obdržálek
  • Peter Rossmanith
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5917)


In contrast to undirected width measures (such as tree-width or clique-width), which have provided many important algorithmic applications, analogous measures for digraphs such as DAG-width or Kelly-width do not seem so successful. Several recent papers, e.g. those of Kreutzer–Ordyniak, Dankelmann–Gutin–Kim, or Lampis–Kaouri–Mitsou, have given some evidence for this. We support this direction by showing that many quite different problems remain hard even on graph classes that are restricted very beyond simply having small DAG-width. To this end, we introduce new measures K-width and DAG-depth. On the positive side, we also note that taking Kanté’s directed generalization of rank-width as a parameter makes many problems fixed parameter tractable.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Bar06]
    Barát, J.: Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics 22(2), 161–172 (2006)CrossRefMATHMathSciNetGoogle Scholar
  2. [BDHK06]
    Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S.: DAG-width and parity games. In: Durand, B., Thomas, W. (eds.) STACS 2006. LNCS, vol. 3884, pp. 524–536. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. [BG04]
    Berwanger, D., Grädel, E.: Entanglement – a measure for the complexity of directed graphs with applications to logic and games. In: Baader, F., Voronkov, A. (eds.) LPAR 2004. LNCS (LNAI), vol. 3452, pp. 209–223. Springer, Heidelberg (2005)Google Scholar
  4. [BJK09]
    Bang-Jensen, J., Kriesell, M.: Disjoint directed and undirected paths and cycles in digraphs. Technical Report PP-2009-03, University of South Denmark (2009)Google Scholar
  5. [BK08]
    Baier, C., Katoen, J.-P.: Principles of Model Checking. The MIT Press, Cambridge (2008)MATHGoogle Scholar
  6. [BXTV08]
    Bui-Xuan, B.-M., Telle, J., Vatshelle, M.: H-join and algorithms on graphs of bounded rank-width (submitted) (November 2008)Google Scholar
  7. [CD06]
    Culus, J.-F., Demange, M.: Oriented coloring: Complexity and approximation. In: Wiedermann, J., Tel, G., Pokorný, J., Bieliková, M., Štuller, J. (eds.) SOFSEM 2006. LNCS, vol. 3831, pp. 226–236. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  8. [CK07]
    Courcelle, B., Kanté, M.: Graph operations characterizing rank-width and balanced graph expressions. In: Brandstädt, A., Kratsch, D., Müller, H. (eds.) WG 2007. LNCS, vol. 4769, pp. 66–75. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  9. [CKL+09]
    Chen, J., Kneis, J., Lu, S., Mölle, D., Richter, S., Rossmanith, P., Sze, S., Zhang, F.: Randomized divide-and-conquer: Improved path, matching, and packing algorithms. SIAM Journal on Computing 38(6), 2526–2547 (2009)MathSciNetGoogle Scholar
  10. [CMR00]
    Courcelle, B., Makowsky, J.A., Rotics, U.: Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst. 33(2), 125–150 (2000)CrossRefMATHMathSciNetGoogle Scholar
  11. [DGK08]
    Dankelmann, P., Gutin, G., Kim, E.: On complexity of minimun leaf out-branching. arXiv:0808.0980v1 (August 2008)Google Scholar
  12. [Egg63]
    Eggan, L.: Transition graphs and the star-height of regular events. Michigan Mathematical Journal 10(4), 385–397 (1963)CrossRefMATHMathSciNetGoogle Scholar
  13. [FGLS09]
    Fomin, F., Golovach, P., Lokshtanov, D., Saurab, S.: Clique-width: On the price of generality. In: SODA 2009, pp. 825–834. SIAM, Philadelphia (2009)Google Scholar
  14. [FHW80]
    Fortune, S., Hopcroft, J.E., Wyllie, J.: The directed subgraph homeomorphism problem. Theor. Comput. Sci. 10, 111–121 (1980)CrossRefMATHMathSciNetGoogle Scholar
  15. [GH08]
    Ganian, R., Hliněný, P.: Automata approach to graphs of bounded rank-width. In: IWOCA 2008, pp. 4–15 (2008)Google Scholar
  16. [GH09]
    Ganian, R., Hliněný, P.: Better polynomial algorithms on graphs of bounded rank-width. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 266–277. Springer, Heidelberg (2009)Google Scholar
  17. [GJ79]
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-completeness. W.H. Freeman, New York (1979)MATHGoogle Scholar
  18. [Gru08]
    Gruber, H.: Digraph complexity measures and applications in formal language theory. In: MEMICS 2008, pp. 60–67 (2008)Google Scholar
  19. [GTW02]
    Grädel, E., Thomas, W., Wilke, T. (eds.): Automata, Logics, and Infinite Games. LNCS, vol. 2500. Springer, Heidelberg (2002)MATHGoogle Scholar
  20. [GW06]
    Gurski, F., Wanke, E.: Vertex disjoint paths on clique-width bounded graphs. Theor. Comput. Sci. 359(1-3), 188–199 (2006)CrossRefMATHMathSciNetGoogle Scholar
  21. [HK08]
    Hunter, P., Kreutzer, S.: Digraph measures: Kelly decompositions, games, and orderings. Theor. Comput. Sci. 399(3), 206–219 (2008)CrossRefMATHMathSciNetGoogle Scholar
  22. [HO06]
    Hliněný, P., Obdržálek, J.: Escape-width: Measuring ”width” of digraphs. Presented at Sixth Czech-Slovak International Symposium on Combinatorics, Graph Theory, Algorithms and Applications (2006)Google Scholar
  23. [HO08]
    Hliněný, P., Oum, S.: Finding branch-decomposition and rank-decomposition. SIAM J. Comput. 38, 1012–1032 (2008)CrossRefMATHMathSciNetGoogle Scholar
  24. [HRW92]
    Hwang, F., Richards, D., Winter, P.: The Steiner Tree Problem. Annals of Discrete Mathematics. North-Holland, Amsterdam (1992)MATHGoogle Scholar
  25. [JRST01]
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory, Series B 82(1), 138–154 (2001)CrossRefMATHMathSciNetGoogle Scholar
  26. [Kan08]
    Kanté, M.: The rank-width of directed graphs. arXiv:0709.1433v3 (March 2008)Google Scholar
  27. [KM04]
    Klostermeyer, W., MacGillivray, G.: Homomorphisms and oriented colorings of equivalence classes of oriented graphs. Discrete Mathematics 274, 161–172 (2004)CrossRefMATHMathSciNetGoogle Scholar
  28. [KO08]
    Kreutzer, S., Ordyniak, S.: Digraph decompositions and monotonicity in digraph searching. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 336–347. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  29. [LKM08]
    Lampis, M., Kaouri, G., Mitsou, V.: On the algorithmic effectiveness of digraph decompositions and complexity measures. In: Hong, S.-H., Nagamochi, H., Fukunaga, T. (eds.) ISAAC 2008. LNCS, vol. 5369, pp. 220–231. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  30. [NdM06]
    Nešetřil, J., Ossona de Mendez, P.: Tree-depth, subgraph coloring and homomorphism bounds. European J. Combin. 27(6), 1024–1041 (2006)Google Scholar
  31. [Obd03]
    Obdržálek, J.: Fast mu-calculus model checking when tree-width is bounded. In: Hunt Jr., W.A., Somenzi, F. (eds.) CAV 2003. LNCS, vol. 2725, pp. 80–92. Springer, Heidelberg (2003)Google Scholar
  32. [Obd06]
    Obdržálek, J.: DAG-width – connectivity measure for directed graphs. In: SODA 2006, pp. 814–821. ACM-SIAM, New York (2006)CrossRefGoogle Scholar
  33. [Obd07]
    Obdržálek, J.: Clique-width and parity games. In: Duparc, J., Henzinger, T.A. (eds.) CSL 2007. LNCS, vol. 4646, pp. 54–68. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  34. [RS86]
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms 7(3), 309–322 (1986)CrossRefMATHMathSciNetGoogle Scholar
  35. [RS91]
    Robertson, N., Seymour, P.D.: Graph minors. X. Obstructions to tree-decomposition. J. Comb. Theory B 52(2), 153–190 (1991)CrossRefMATHMathSciNetGoogle Scholar
  36. [Saf05]
    Safari, M.: D-width: A more natural measure for directed tree-width. In: Jedrzejowicz, J., Szepietowski, A. (eds.) MFCS 2005. LNCS, vol. 3618, pp. 745–756. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  37. [vL76]
    van Leeuwen, J.: Having a Grundy-numbering is NP-complete. Technical Report 207, The Pennsylvania State University (September 1976)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Robert Ganian
    • 1
  • Petr Hliněný
    • 1
  • Joachim Kneis
    • 2
  • Alexander Langer
    • 2
  • Jan Obdržálek
    • 1
  • Peter Rossmanith
    • 2
  1. 1.Faculty of InformaticsMasaryk UniversityBrnoCzech Republic
  2. 2.Theoretical Computer ScienceRWTH Aachen UniversityGermany

Personalised recommendations