Abstract
A graph is said to be prime if it has no non-trivial substitution decomposition, or module. This paper introduces a simple but efficient (O(n + m ln n)) algorithm to test the primality of undirected graphs. The fastest previous algorithm is due to Muller and Spinrad [MS89] and requires quadratic time. Our approach can be seen as a common generalization of Spinrad's work on P 4-tree structure and substitution decomposition [Spi89] and Ille's one about the structure of prime graphs [Ill90] (see also Schmerl and Trotter [ST91] which contains similar results).
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References
M. Chein, M. Habib, and M. C. Maurer. Partitive hypergraphs. Discrete Mathematics, (37):35–50, 1981.
D. G. Corneil, Y. Perl, and L. K. Stewart. A linear recognition algorithm for cographs. SIAM journal of computing, (14):926–934, 1985.
M. Habib. Substitution des structures combinatoires, théorie et algorithmes, thèse d'état, Paris VI, 1981.
M. Habib and M. C. Maurer. On the X-join decomposition for undirected graphs. J. Discrete Applied Math, (3):198–207, 1979.
P. Ille. Indecomposable relations. preprint Université de Marseille, 1990.
P. Ille. L'ensemble des intervalles d'une multirelation binaire et reflexive. Zeitschr. f. math. Logik und Grundlagen d. Math. Bd., (37):227–256, 1991.
D. Kelly. Graphs and Orders, chapter Comparability graphs, pages 3–40. D. Reidel Publishing Company, 1985.
R. H. Möring and F. J. Radermacher. Substitution decomposition for discrete structures and connections with combinatorial optimization. Ann. Discrete math, (19):257–356, 1984.
J. H. Muller and J. Spinrad. Incremental modular decomposition. Journal of the association for Computing Machinery, (1):1–19, January 1989.
J. Spinrad. P4-trees and substitution decomposition. preprint, Vanderbild U., Nashville, 1989.
J. H. Schmerl and W. T. Trotter. Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures. preprint, 1991.
D. P. Sumner. Graphs indecomposable with respect to the X-join. Discrete Mathematics, 6:2811–298, 1973.
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© 1993 Springer-Verlag Berlin Heidelberg
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Alain, C., Michel, H. (1993). An efficient algorithm to recognize prime undirected graphs. In: Mayr, E.W. (eds) Graph-Theoretic Concepts in Computer Science. WG 1992. Lecture Notes in Computer Science, vol 657. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56402-0_49
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DOI: https://doi.org/10.1007/3-540-56402-0_49
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