Abstract
This paper gives anO(n 2) incremental algorithm for computing the modular decomposition of 2-structures [1], [2]. A 2-structure is a type of edge-colored graph, and its modular decomposition is also known as the prime tree family. Modular decomposition of 2-structures arises in the study of relational systems. The modular decomposition of undirected graphs and digraphs is a special case, and has applications in a number of combinatorial optimization problems. This algorithm generalizes elements of a previousO(n 2) algorithm of Muller and Spinrad [3] for the decomposition of undirected graphs. However, Muller and Spinrad's algorithm employs a sophisticated data structure that impedes its generalization to digraphs and 2-structures, and limits its practical use. We replace this data structure with a scheme that labels each edge with at most one node, thereby obtaining an algorithm that is both practical and general to 2-structures.
Similar content being viewed by others
References
A. Ehrenfeucht and G. Rozenberg, Theory of 2-structures, part 1: Clans, basic subclasses, and morphisms,Theoretical Computer Science,70 (1990), 277–303.
A. Ehrenfeucht and G. Rosenberg, Theory of 2-structures, part 2: Representations through labeled tree families,Theoretical Computer Science,70 (1990), 305–342.
J. H. Muller and J. Spinrad, Incremental modular decomposition,Journal of the Association for Computing Machinery,36 (1989), 1–19.
R. H. Möhring and F. J. Radermacher, Substitution decomposition for discrete structures and connections with combinatorial optimization,Annals of Discrete Mathematics,19 (1984), 257–356.
R. H. Möhring, Algorithmic aspects of the substitution decomposition in optimization over relations, set systems and boolean functions,Annals of Operation Research,4 (1985/6), 195–225.
R. H. Möhring, Algorithmic aspects of comparability graphs and interval graphs, inGraphs and Orders (I. Rival, editor), Reidel, Boston, 1985, pp. 41–101.
J. H. Schmerl, Arborescent structures, II: interpretability in the theory of trees,Transactions of the American Mathematical Society,266 (1981), 629–643.
T. Gallai, Transitiv orientierbare graphen,Acta Mathematica Acadamiae Scientiarum Hungaricae,18 (1967), 25–66.
D. Kelly, Comparability graphs, inGraphs and Order (I. Rival, editor), Reidel, Boston, 1985, pp. 3–40.
M. Habib and M. C. Maurer, On the X-join decomposition for undirected graphs,Discrete Applied Mathematics,1 (1979), 201–207.
M. C. Golumbic,Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
A. Blass, Graphs with unique maximal dumpings,Journal of Graph Theory,2 (1978), 19–24.
L. N. Shevrin and N. D. Filippov, Partially ordered sets and their comparability graphs,Siberian Mathematical Journal,11 (1970), 497–509.
D. G. Corneil, Y. Perl, and L. K. Stewart, A linear recognition algorithm for cographs,SIAM Journal on Algebraic and Discrete Methods,3 (1985), 926–934.
J. Valdes, R. E. Tarjan, and E. L. Lawler, The recognition of series-parallel digraphs,SIAM Journal on Computing,11 (1982), 299–313.
A. Ehrenfeucht and R. M. McConnell, Ak-structure generalization of the theory of 2-structures,Theoretical Computer Science,132 (1994), 209–227.
W. H. Cunningham and J. Edmonds, A combinatorial decomposition theory,Canadian Journal of Mathematics,32 (1980), 734–765.
D. Wagner, Decomposition of k-ary relations,Discrete Mathematics,81 (1990), 303–322.
L. O. James, R. G. Stanton, and D. D. Cowan, Graph decomposition for undirected graphs, inProceedings of the 3rd South-Eastern Conference on Combinatorics, Graph Theory and Computing (F. Hoffman and R. B. Levow, editors), Utilitas Mathematica, Winnipeg, 1972, pp. 281–290.
B. Buer and R. H. Möhring, A fast algorithm for the decomposition of graphs and posets,Mathematics of Operations Research,8 (1983), 170–184.
M. C. Golumbic, Comparability graphs and a new matroid,Journal of Combinatorial Theory, Series B,22 (1977), 68–90.
W. H. Cunningham, Decomposition of directed graphs,SIAM Journal of Algebraic and Discrete Methods,3 (1982), 214–228.
G. Steiner, Machine Scheduling with Precedence Constraints, Ph.D. thesis, University of Waterloo, Waterloo, Ontario, 1982.
C. L. McCreary, An Algorithm for Parsing a Graph Grammar, Ph.D. thesis, University of Colorado, Boulder, Co, 1987.
A. Cournier and M. Habib, An Efficient Algorithm To Recognize Prime Undirected Graphs, Technical Report R.R. LIRMM 92-023, Laboratoire D'Informatique, de Robotique et de Microelectronique de Montpellier, 1992.
J. P. Spinrad,P 4 trees and substitution decomposition,Discrete Applied Mathematics,39 (1992), 263–291.
R. M. McConnell and J. P. Spinrad, Linear-time modular decomposition of undirected graphs and efficient transitive orientation of comparability graphs,Proceedings of the Symposium on Discrete Algorithms, 1993, pp. 536–545.
M. C. Maurer, Joints et decompositions premiéres dans les graphes, Thèse 3ème cycle, Université de Paris VI, 1977.
A. Ehrenfeucht, T. Harju, and G. Rozenberg, Incremental construction of 2-structures,Discrete Mathematics,128 (1994), 113–141.
A. Ehrenfeucht, H. N. Gabow, R. M. McCeonnell, and S. J. Sullivan, AnO(n 2) algorithm to compute the prime tree family of a 2-structure,Journal of Algorithms,16 (1994), 283–294.
M. Chein, M. Habib, and M. C. Maurer, Partitive hypergraphs,Discrete Mathematics,37 (1981), 35–50.
J. H. Schmerl and W. T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures,Discrete Mathematics,113 (1993), 191–205.
Author information
Authors and Affiliations
Additional information
Communicated by K. Mehlhorn.
Rights and permissions
About this article
Cite this article
McConnell, R.M. AnO(n 2) incremental algorithm for modular decomposition of graphs and 2-structures. Algorithmica 14, 229–248 (1995). https://doi.org/10.1007/BF01206330
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01206330