DAG-Width and Parity Games

  • Dietmar Berwanger
  • Anuj Dawar
  • Paul Hunter
  • Stephan Kreutzer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 3884)


Tree-width is a well-known metric on undirected graphs that measures how tree-like a graph is and gives a notion of graph decomposition that proves useful in algorithm development. Tree-width is characterised by a game known as the cops-and-robber game where a number of cops chase a robber on the graph. We consider the natural adaptation of this game to directed graphs and show that monotone strategies in the game yield a measure with an associated notion of graph decomposition that can be seen to describe how close a directed graph is to a directed acyclic graph (DAG). This promises to be useful in developing algorithms on directed graphs. In particular, we show that the problem of determining the winner of a parity game is solvable in polynomial time on graphs of bounded DAG-width. We also consider the relationship between DAG-width and other measures such as entanglement and directed tree-width. One consequence we obtain is that certain NP-complete problems such as Hamiltonicity and disjoint paths are polynomial-time computable on graphs of bounded DAG-width.


Polynomial Time Directed Graph Directed Acyclic Graph Undirected Graph Polynomial Time Algorithm 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    J. Barát, Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics (to appear)Google Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: Treewidth: Algorithmic techniques and results. In: Privara, I., Ružička, P. (eds.) MFCS 1997. LNCS, vol. 1295, pp. 19–36. Springer, Heidelberg (1997)CrossRefGoogle Scholar
  4. 4.
    Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: van Leeuwan, J. (ed.) Handbook of Theoretical Computer Science. Formal Models and Sematics (B), vol. B, pp. 193–242 (1990)Google Scholar
  5. 5.
    Dendris, N.D., Kirousis, L.M., Thilikos, D.M.: Fugitive-search games on graphs and related parameters. TCS 172, 233–254 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Emerson, E., Jutla, C., Sistla, A.: On model checking for the μ-calculus and its fragments. TCS 258, 491–522 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gottlob, G., Leone, N., Scarcello, F.: Robbers, marshals, and guards: Game theoretic and logical characterizations of hypertree width. In: PODS, pp. 195–201 (2001)Google Scholar
  8. 8.
    Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory, Series B 82, 138–154 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Jurdziński, M.: Deciding the winner in parity games is in UP ∩ co-UP. Information Processing Letters 68, 119–124 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kozen, D.: Results on the propositional mu-calculus. TCS 27, 333–354 (1983)CrossRefzbMATHGoogle Scholar
  11. 11.
    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)CrossRefGoogle Scholar
  12. 12.
    Reed, B.A.: Introducing directed tree width. In: 6th Twente Workshop on Graphs and Combinatorial Optimization. Electron. Notes Discrete Math., vol. 3, Elsevier, Amsterdam (1999)Google Scholar
  13. 13.
    Robertson, N., Seymour, P.: Graph Minors. III. Planar tree-width. Journal of Combinatorial Theory, Series B 36, 49–63 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    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
  15. 15.
    Seymour, P., Thomas, R.: Graph searching, and a min-max theorem for tree-width. Journal of Combinatorial Theory, Series B 58, 22–33 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Dietmar Berwanger
    • 1
  • Anuj Dawar
    • 2
  • Paul Hunter
    • 3
  • Stephan Kreutzer
    • 3
  1. 1.LaBRI, Université de Bordeaux 1France
  2. 2.University of Cambridge Computer LaboratoryUK
  3. 3.Logic and Discrete Systems, Institute for Computer ScienceHumboldt-University BerlinGermany

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