# Better algorithms for the pathwidth and treewidth of graphs

Algorithms (Session 13)

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## Abstract

In this paper we give, for all constants *k*, explicit *O*(*n* log^{2}*n*) algorithms, that given a graph *G* = (*V,E*), decide whether the treewidth (or pathwidth) of *G* is at most *k*, and if so, find a tree-decomposition or (path-decomposition) of *G* with treewidth (or pathwidth) at most *k*. In contrast with previous solutions, our algorithms do not rely on non-constructive reasoning, and are single exponential in *k*. This result implies a similar result for several graph notions that are equivalent with treewidth or pathwidth.

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## References

- [1]S. Arnborg. Efficient algorithms for combinatorial problems on graphs with bounded decomposability — A survey.
*BIT*, 25:2–23, 1985.Google Scholar - [2]S. Arnborg, D. G. Corneil, and A. Proskurowski. Complexity of finding embeddings in a
*k*-tree.*SIAM J. Alg. Disc. Meth.*, 8:277–284, 1987.Google Scholar - [3]S. Arnborg, B. Courcelle, A. Proskurowski, and D. Seese. An algebraic theory of graph reduction. Technical Report 90-02, Laboratoire Bordelais de Recherche en Informatique, Bordeaux, 1990. To appear in Proceedings 4th Workshop on Graph Grammars and Their Applications to Computer Science.Google Scholar
- [4]S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs (extended abstract). In
*Proceedings of the 15'th International Colloquium on Automata, Languages and Programming*, pages 38–51. Springer Verlag, Lect. Notes in Comp. Sc. 317, 1988. To appear in J. of Algorithms.Google Scholar - [5]S. Arnborg and A. Proskurowski. Characterization and recognition of partial 3-trees.
*SIAM J. Alg. Disc. Meth.*, 7:305–314, 1986.Google Scholar - [6]S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial
*k*-trees.*Disc. Appl. Math.*, 23:11–24, 1989.Google Scholar - [7]M. W. Bern, E. L. Lawler, and A. L. Wong. Linear time computation of optimal subgraphs of decomposable graphs.
*J. Algorithms*, 8:216–235, 1987.Google Scholar - [8]H. L. Bodlaender. Classes of graphs with bounded treewidth. Technical Report RUU-CS-86-22, Dept. of Computer Science, Utrecht University, Utrecht, 1986.Google Scholar
- [9]H. L. Bodlaender. Dynamic programming algorithms on graphs with bounded tree-width. In
*Proceedings of the 15'th International Colloquium on Automata, Languages and Programming*, pages 105–119. Springer Verlag, Lecture Notes in Computer Science, vol. 317, 1988.Google Scholar - [10]H. L. Bodlaender. NC-algorithms for graphs with small treewidth. In J. van Leeuwen, editor,
*Proc. Workshop on Graph-Theoretic Concepts in Computer Science WG'88*, pages 1–10. Springer Verlag, LNCS 344, 1988.Google Scholar - [11]H. L. Bodlaender. Complexity of path forming games. Technical Report RUU-CS-89-29, Utrecht University, Utrecht, 1989.Google Scholar
- [12]H. L. Bodlaender. Improved self-reduction algorithms for graphs with bounded treewidth. In
*Proc. 15th Int. Workshop on Graph-theoretic Concepts in Computer Science WG'89*, pages 232–244. Springer Verlag, Lect. Notes in Computer Science, vol. 411, 1990. To appear in: Annals of Discrete Mathematics.Google Scholar - [13]R. B. Borie, R. G. Parker, and C. A. Tovey. Automatic generation of linear algorithms from predicate calculus descriptions of problems on recursive constructed graph families. Manuscript, 1988.Google Scholar
- [14]N. Chandrasekharan and S. T. Hedetniemi. Fast parallel algorithms for tree decomposing and parsing partial
*k*-trees. In*Proc. 26th Annual Allerton Conference on Communication, Control, and Computing*, Urbana-Champaign, Illinois, 1988.Google Scholar - [15]B. Courcelle. The monadic second-order logic of graphs I: Recognizable sets of finite graphs.
*Information and Computation*, 85:12–75, 1990.Google Scholar - [16]I. S. Duff and J. K. Reid. The multifrontal solution of indefinite sparse symmetric linear equations.
*ACM Transactions on Mathematical Software*, 9:302–325, 1983.Google Scholar - [17]J. A. Ellis, I. H. Sudborough, and J. Turner. Graph separation and search number. Report DCS-66-IR, University of Victoria, 1987.Google Scholar
- [18]M. R. Fellows and K. R. Abrahamson. Cutset regularity beats well-quasi-ordering for bounded treewidth. Manuscript, 1990.Google Scholar
- [19]M. R. Fellows and M. A. Langston. An analogue of the Myhill-Nerode theorem and its use in computing finite-basis characterizations. In
*Proceedings of the 30th Annual Symposium on Foundations of Computer Science*, pages 520–525, 1989.Google Scholar - [20]M. R. Fellows and M. A. Langston. On search, decision and the efficiency of polynomial-time algorithms. In
*Proceedings of the 21th Annual Symposium on Theory of Computing*, pages 501–512, 1989.Google Scholar - [21]A. Habel.
*Hyperedge Replacement: Grammars and Languages*. PhD thesis, Univ. Bremen, 1988.Google Scholar - [22]J. Lagergren. Efficient parallel algorithms for tree-decomposition and related problems. In
*Proceedings of the 31th Annual Symposium on Foundations of Computer Science*, pages 173–182, 1990.Google Scholar - [23]J. Lagergren.
*Algorithms and Minimal Forbidden Minors for Tree-decomposable Graphs*. PhD thesis, Royal Institute of Technology, Stockholm, Sweden, 1991.Google Scholar - [24]J. Lagergren and S. Arnborg. Finding minimal forbidden minors using a finite congruence. To appear in: proceedings ICALP'91.Google Scholar
- [25]S. Lauritzen and D. Spiegelhalter. Local computations with probabilities on graphical structures and their application to expert systems.
*The Journal of the Royal Statistical Society. Series B (Methodological)*, 50:157–224, 1988.Google Scholar - [26]J. Matousěk and R. Thomas. Algorithms finding tree-decompositions of graphs.
*J. Algorithms*, 12:1–22, 1991.Google Scholar - [27]R. H. Möhring. Graph problems related to gate matrix layout and PLA folding. Technical Report 223/1989, Technical University of Berlin, 1989.Google Scholar
- [28]N. Robertson and P. D. Seymour. Graph minors — a survey. In I. Anderson, editor,
*Surveys in Combinatorics*, pages 153–171. Cambridge Univ. Press, 1985.Google Scholar - [29]P. Scheffler. Linear-time algorithms for NP-complete problems restricted to partial k-trees. Report R-MATH-03/87, Karl-Weierstrass-Institut Für Mathematik, Berlin, GDR, 1987.Google Scholar
- [30]L. C. van der Gaag.
*Probability-Based Models for Plausible Reasoning*. PhD thesis, University of Amsterdam, 1990.Google Scholar - [31]T. V. Wimer.
*Linear algorithms on k-terminal graphs*. PhD thesis, Dept. of Computer Science, Clemson University, 1987.Google Scholar

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© Springer-Verlag Berlin Heidelberg 1991