Abstract
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.
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References
J. Barát, Directed path-width and monotonicity in digraph searching. Graphs and Combinatorics (to appear)
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)
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)
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)
Dendris, N.D., Kirousis, L.M., Thilikos, D.M.: Fugitive-search games on graphs and related parameters. TCS 172, 233–254 (1997)
Emerson, E., Jutla, C., Sistla, A.: On model checking for the μ-calculus and its fragments. TCS 258, 491–522 (2001)
Gottlob, G., Leone, N., Scarcello, F.: Robbers, marshals, and guards: Game theoretic and logical characterizations of hypertree width. In: PODS, pp. 195–201 (2001)
Johnson, T., Robertson, N., Seymour, P.D., Thomas, R.: Directed tree-width. Journal of Combinatorial Theory, Series B 82, 138–154 (2001)
Jurdziński, M.: Deciding the winner in parity games is in UP ∩ co-UP. Information Processing Letters 68, 119–124 (1998)
Kozen, D.: Results on the propositional mu-calculus. TCS 27, 333–354 (1983)
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)
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)
Robertson, N., Seymour, P.: Graph Minors. III. Planar tree-width. Journal of Combinatorial Theory, Series B 36, 49–63 (1984)
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)
Seymour, P., Thomas, R.: Graph searching, and a min-max theorem for tree-width. Journal of Combinatorial Theory, Series B 58, 22–33 (1993)
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Berwanger, D., Dawar, A., Hunter, P., Kreutzer, S. (2006). DAG-Width and Parity Games. In: Durand, B., Thomas, W. (eds) STACS 2006. STACS 2006. Lecture Notes in Computer Science, vol 3884. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11672142_43
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DOI: https://doi.org/10.1007/11672142_43
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