# Approximating treewidth and pathwidth of some classes of perfect graphs

## Abstract

In this paper we discuss algorithms that approximate the treewidth and pathwidth of cotriangulated graphs, permutation graphs and of cocomparability graphs. For a cotriangulated graph, of which the treewidth is at most *k* we show there exists an *O*(n^{2}) algorithm finding a path-decomposition with width at most 3*k*+4. If *G[π]* is a permutation graph with treewidth *k*, then we show that the pathwidth of *G[π]* is at most 2*k*, and we give an algorithm which constructs a path-decomposition with width at most 2*k* in time *O(nk)*. We assume that the permutation *π* is given. In this paper we also discuss the problem of finding an approximation for the treewidth and pathwidth of cocomparability graphs. We show that, if the treewidth of a cocomparability graph is at most *k*, then the pathwidth is at most *O(k*^{2}), and we give a simple algorithm finding a path-decomposition with this width. The running time of the algorithm is dominated by a coloring algorithm of the graph. Such a coloring can be found in time *O(n*^{3}).

If the treewidth is bounded by some constant, previous results (i.e. [10, 21]), show that, once the approximations are given, the exact treewidth and pathwidth can be computed in linear time for all these graphs.

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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, J. Lagergren and D. Seese, Easy problems for tree-decomposable graphs,
*J. Algorithms***12**, 308–340, 1991.CrossRefGoogle Scholar - 4.S. Arnborg and A. Proskurowski, Characterization and recognition of partial 3-trees,
*SIAM J. Alg. Disc. Meth.***7**, 305–314, 1986.Google Scholar - 5.S. Arnborg and A. Proskurowski, Linear time algorithms for NP-hard problems restricted to partial
*k*-trees.*Disc. Appl. Math.***23**, 11–24, 1989.CrossRefGoogle Scholar - 6.
- 7.H.L. Bodlaender, A tourist guide through treewidth, Technical report RUU-CS-92-12, Department of computer science, Utrecht University, Utrecht, The Netherlands, 1992. To appear in:
*Proceedings 7th International Meeting of Young Computer Scientists*, Springer Verlag, Lecture Notes in Computer Science.Google Scholar - 8.H. Bodlaender, J. Gilbert, H. Hafsteinsson and T. Kloks, Approximating treewidth, pathwidth and minimum elimination tree height, In G. Schmidt and R. Berghammer, editors,
*Proceedings 17th International Workshop on Graph-Theoretic Concepts in Computer Science WG'91*, 1–12, Springer Verlag, Lecture Notes in Computer Science, vol. 570, 1992.Google Scholar - 9.H. Bodlaender and R.H. Möhring, The pathwidth and treewidth of cographs, In
*Proceedings 2nd Scandinavian Workshop on Algorithm Theory*, 301–309, Springer Verlag, Lecture Notes in Computer Science vol. 447, 1990.Google Scholar - 10.H. Bodlaender and T. Kloks, Better algorithms for the pathwidth and treewidth of graphs,
*Proceedings of the 18th International colloquium on Automata, Languages and Programming*, 544–555, Springer Verlag, Lecture Notes in Computer Science, vol. 510, 1991.Google Scholar - 11.A. BrandstÄdt and D. Kratsch, On the restriction of some NP-complete graph problems to permutation graphs,
*Fundamentals of Computation Theory, proc. FCT*1985, 53–62, Lecture Notes in Comp. Science vol. 199, Springer Verlag, New York, 1985.Google Scholar - 12.A. BrandstÄdt and D. Kratsch, On domination problems for permutation and other perfect graphs,
*Theor. Comput. Sci.***54**, 181–198, 1987.CrossRefGoogle Scholar - 13.B. Courcelle, The monadic second-order logic of graphs I: Recognizable sets of finite graphs,
*Information and Computation***85**, 12–75, 1990.CrossRefGoogle Scholar - 14.B. Courcelle, The monadic second-order logic of graphs III: Treewidth, forbidden minors and complexity issues, Report 8852, University Bordeaux 1, 1988.Google Scholar
- 15.B. Courcelle, Graph rewriting: an algebraic and logical approach. In J. van Leeuwen, editor,
*Handbook of Theoretical Computer Science, Vol. B*, 192–242, Amsterdam, 1990. North Holland Publ. Comp.Google Scholar - 16.
- 17.S. Even, A. Pnueli and A. Lempel, Permutation graphs and transitive graphs,
*J. Assoc. Comput. Mach.***19**, 400–410, 1972.Google Scholar - 18.Gilmore and Hoffman, A characterization of comparability and interval graphs,
*Canad. J. Math.***16**, 539–548, 1964.Google Scholar - 19.M.C. Golumbic,
*Algorithmic Graph Theory and Perfect Graphs*, Academic Press, New York, 1980.Google Scholar - 20.J. Gustedt, Pathwidth for chordal graphs is NP-complete. Preprint TU Berlin, 1989.Google Scholar
- 21.J. Lagergren and S. Arnborg, Finding minimal forbidden minors using a finite congruence,
*Proceedings of the 18th International colloquium on Automata, Languages and Programming*, 532–543, Springer Verlag, Lecture Notes in Computer Science, vol. 510, 1991.Google Scholar - 22.J. van Leeuwen, Graph algorithms. In
*Handbook of Theoretical Computer Science, A: Algorithms an Complexity Theory*, 527–631, Amsterdam, 1990. North Holland Publ. Comp.Google Scholar - 23.J. Matoušek and R. Thomas, Algorithms Finding Tree-Decompositions of Graphs,
*Journal of Algorithms***12**, 1–22, 1991.CrossRefGoogle Scholar - 24.A. Pnueli, A. Lempel, and S. Even, Transitive orientation of graphs and identification of permutation graphs,
*Canad. J. Math.***23**, 160–175, 1971.Google Scholar - 25.B. Reed, Finding approximate separators and computing treewidth quickly, To appear in: Proceedings STOC'92, 1992.Google Scholar
- 26.N. Robertson and P.D. Seymour, Graph minors—A survey. In I. Anderson, editor,
*Surveys in Combinatorics*, 153–171. Cambridge Univ. Press 1985.Google Scholar - 27.N. Robertson and P.D. Seymour, Graph minors II. Algorithmic aspects of treewidth.
*J. Algorithms***7**, 309–322, 1986.CrossRefGoogle Scholar - 28.F.S. Roberts, Graph theory and its applications to problems of society, NFS-CBMS Monograph no. 29 (SIAM Publications, Philadelphia, PA. 1978).Google Scholar
- 29.M.C. Golumbic, D. Rotem and J. Urrutia, Comparability graphs and intersection graphs,
*Discrete Math.***43**, 37–46, 1983.CrossRefGoogle Scholar - 30.J. Spinrad, On comparability and permutation graphs,
*SIAM J. Comp.***14**, No. 3, August 1985.Google Scholar - 31.R. Sundaram, K. Sher Singh and C. Pandu Rangan, Treewidth of circular arc graphs, Manuscript 1991.Google Scholar