Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-04T21:35:13.072Z Has data issue: false hasContentIssue false
  • English
  • Français

Decomposable graphs and hypergraphs

Part of: Graph theory

Published online by Cambridge University Press:  09 April 2009

S. L. Lauritzen ,
T. P. Speed  and
K. Vijayan
S. L. Lauritzen
Affiliation:
Institute for Elektronik Systems Aalborg UniversityBadehusvej 13 Aalborg, DK-9000, Denmark
T. P. Speed
Affiliation:
CSIRO Division of Mathematics and Statistics P.O. Box 1965, Canberra City ACT, 2601, Australia
K. Vijayan
Affiliation:
Department of Mathematics University of Western AustraliaNedlands, W.A. 6009, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We define and investigate the notion of a decomposable hypergraph, showing that such a hypergraph always is conformal, that is, can be viewed as the class of maximal cliques of a graph. We further show that the clique hypergraph of a graph is decomposable if and only if the graph is triangulated and characterise such graphs in terms of a combinatorial identity.

Keywords

clique hypergraphstriangulated graphs.

MSC classification

Secondary: 05C65: Hypergraphs 05C10: Planar graphs; geometric and topological aspects of graph theory
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 36 , Issue 1 , February 1984 , pp. 12 - 29
Copyright
Copyright © Australian Mathematical Society 1984

References

Anderson, A. H. (1974), ‘Multidimensional contingency tables,’ Scand. J. Statist. 1, 115–127.Google Scholar
Berge, C. (1967), ‘Some classes of perfect graphs’, Graph theory and theoretical physics, edited by Harary, F. (Academic Press, New York, London).Google Scholar
Berge, C. (1973). Graphs and hypergraphs, translated from French by Edward, Minieka (North-Holland Publishing Company, Amsterdam, London; American Elsevier Publishing Inc., New York).Google Scholar
Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975), Discrete multivariate analysis: theory and practice (MIT Press, Cambridge, Massachusetts and London, England).Google Scholar
Dirac, G. A. (1961), ‘On rigid circuit graphs,’ Abh. Math. Sem. Univ. Hamburg. 25, 71–76.Google Scholar
Fulkerson, D. R. and Gross, O. A. (1965), ‘Incidence matrices and interval graphs,’ Pacific J. Math. 15, 835–855.Google Scholar
Gavril, F. (1972), ‘Algorithms for minimum coloring, maximum clique, minimum covering by cliques and maximum independent set of a chordal graph,’ SIAM J. Comput. 1, 180–187.Google Scholar
Gavril, F. (1974), ‘The intersection graphs of subtrces in trees are exactly the chordal graphs,’ J. Combinatorial Theory 16, 47–56.Google Scholar
Gavril, F. (1975), ‘An algorithm for testing chordality of graphs,’ Information Processing Lett. 3, 110–112.Google Scholar
Golumbic, M. C. (1980), Algorithmic graph theory and perfect graphs (Academic Press, New York).Google Scholar
Goodman, L. A. (1971), ‘Partitioning of chi-square, analysis of marginal contingency tables, and estimation of expected frequencies in multi-dimensional contingency tables,’ J. Amer. Statist. Assoc. 66, 339–344.Google Scholar
Haberman, S. J. (1974), The analysis of frequency data (University of Chicago Press, Chicago, London).Google Scholar
Hajnal, A. and Suranyi, J. (1958), ‘Über die Auflösung von Graphen in vollständige Teilgraphen,’ Ann. Univ. Sci. Budapest. Eötvös Sect. Math. 1, 113–121.Google Scholar
Harary, F. (1969), Graph theory (Addison-Wesley Publishing Company, Reading, Massachusetts, Menlo Park, California, London, Don Mills, Ontario).Google Scholar
Haskins, L. and Rose, D. (1973), ‘Towards the characterization of perfect elimination graphs,’ SIAM J. Comput. 2, 217–224.Google Scholar
Jensen, S. T. (1978), Flersidede kontingenstabeller (Institute of Mathematical Statistics, University of Copenhagen (Danish)).Google Scholar
Kellerer, H. G. (1964), ‘Verteilungsfunktionen mit gegebenen Marginalverteilungen,’ Z. Wahrscheinlichkeitstheorie und Verw. Gehiete 3, 247–270.Google Scholar
Klcitman, D. J. (1974), A note on perfect elimination graphs, SIAM J. Comput. 3. 280–282.Google Scholar
Lekkerkerker, C. G. and Boland, J. C. H. (1962), ‘Representation of a finite graph by a set of intervals on the real line,’ Fund. Math. 51, 45–64.Google Scholar
Lueker, G. S. (1975), Efficient algorithms for chordal graphs and interval graphs (Ph.D. thesis. Princeton University).Google Scholar
Ohtsuki, T. (1976), ‘A fast algorithm for finding an optimal ordering for vertex elimination on a graph,’ SIAM J. Comput. 5, 132–145.Google Scholar
Ohtsuki, T., Cheung, L. I. and Fujisawa, T. (1976), ‘Minimal triangulation of a graph and optimal pivoting order,’ J. Math. Anal. Appl. 54, 622–633.Google Scholar
Parter, S. (1961), ‘The use of linear graphs in Gauss elimination,’ SIAM Review 3, 119–130.Google Scholar
Rose, D. J. (1970), ‘Triangulated graphs and the elimination process,’ J. Math. Anal. Appl. 32, 597–609.Google Scholar
Rose, D. J. (1972), ‘A graph-theoretic study of the numerical solution of sparse positive definite systems of linear equations,’ Graph theory and computing, edited by Read, R. (Academic Press, New York, London, 183–217).Google Scholar
Rose, D. J., Tarjan, R. E. and Leuker, G. S. (1975), ‘Algorithmic aspects of vertex elimination on graphs,’ SIAM J. Comput. 4, 266–283.Google Scholar
Speed, T. P. (1970), ‘Decompositions of graphs and hypergraphs,’ Combinatorial mathematics, edited by Holton, D. A. and Seberry, J. (Lecture Notes in Mathematics 868, pp. 300–307).Google Scholar
Suomela, P. (1976), ‘Construction of nearest-neighbour systems,’ Ann. Acad. Sci. Fenn. Ser A I Math. Dissertationes 10.Google Scholar
Vorob'ev, N. N. (1962), ‘Consistent families of measures and their extensions,’ Theor. Prob. Appl. 7, 147–163.Google Scholar
Vorob'ev, N. N. (1963), ‘Markov measures and Markov extensions,’ Theor. Prob. Appl. 8, 420–429.Google Scholar
Vorob'ev, N. N. (1967), ‘Coalition games,’ Theor. Prob. Appl. 12, 250–266.Google Scholar