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A representation for the modules of a graph and applications

Sulamita Klein^I; Jaime L. Szwarcfiter^II

^IInstituto de Matemática and COPPE/Sistemas, UFRJ, CP 68511,21945-970 - Rio de Janeiro, RJ, Brazil, e-mail: sula@cos.ufrj.br
^IIInstituto de Matemática NCE and COPPE/Sistemas, UFRJ, CP 2324,20001-970 - Rio de Janeiro, RJ, Brazil, e-mail: jayme@nce.ufrj.br

ABSTRACT

We describe a simple representation for the modules of a graph G. We show that the modules of G are in one-to-one correspondence with the ideals of certain posets. These posets are characterized and shown to be layered posets, that is, transitive closures of bipartite tournaments. Additionaly, we describe applications of the representation. Employing the above correspondence, we present methods for solving the following problems: (i) generate all modules of G, (ii) count the number of modules of G, (iii) find a maximal module satisfying some hereditary property of G and (iv) find a connected non-trivial module of G.

Keywords: graphs, ideals, modules, posets, bipartite tournaments, algorithms

References

[1] B. Buer and R. H. Mohring. A fast algorithm for the decomposition of graphs and posets. Math. Oper. Res. 8 170-184, 1983.        [ Links ]

[2] A. Cournier and M. Habib. A new linear algorithm for modular decomposition, in Sophie Tison, ed., CAAP'94: 19th International Colloquium, Lectures Notes in Computer Science Edinburgh, UK, 68-82, 1994.        [ Links ]

[3] E. Dahlhaus, J. Gustedt and R. M. McConnel. Efficient and pratical modular decomposition. Proceedings of the Eight Annual ACM-SIAM Symposium on Discrete Algorithms 26-35, 1997.        [ Links ]

[4] A. Ehrenfeucht, H. N. Gabow, R. M. McConnel and S. J. Sullivan. An O(n²) divide-and-conquer algorithms for the prime tree decomposition of two-structures and modules decompositions of graphs. J. of Algorithms 16 283-294, 1994.        [ Links ]

[5] T. Gallai. Transitiv orientierbare graphen. Acta math. Acad. Sci. hungar. 18, 25-66, 1967.        [ Links ]

[6] M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic press, New York, 1980.        [ Links ]

[7] M. Habib and M. C. Maurer. On the X-join decomposition for undirected graphs. Discrete Appl. Math. 1 201-207, 1979.        [ Links ]

[8] D. Kelly. Comparability graphs, in I. Rival, ed. Graphs and Orders, Reidl, Boston, 3-40, 1985.        [ Links ]

[9] R. M. McConnel and J. Spinrad. Linear-time modular decomposition and efficient transitive orientation of comparability graphs, in Proceedings of the Fifth Annual ACM-SIAM Symposium on Discrete Algorithms 536-545, 1994.        [ Links ]

[10] R. M. McConnel and J. Spinrad. Linear -time transitive orientation, in Proceedings of the Eight Annual ACM-SIAM Symposium on Discrete Algorithms 19-25, 1997.        [ Links ]

[11] R. M. McConnel and J. Spinrad. Modular decomposition and transitive orientation. Available at http://www.cs.amherst.edu/ross/work.html, 1997.        [ Links ]

[12] J. H. Miller and J. Spinrad. Incremental modular decomposition. J. Assoc. Comput. mach. 36 1-19, 1989.        [ Links ]

[13] R. H. Mohring. Algoritmic aspects of comparability graphs and interval graphs, in I. Rival, ed. Graphs and Orders, Reidl, Boston, 41-101, 1985.        [ Links ]

[14] J. Spinrad. P₄ trees and substitution decomposition. Discrete Appl. Math. 39 263-291, 1992.        [ Links ]