Abstract
This paper gives a decomposition theory for bipartite graphs. It uses bimodules, a special case of 2-modules (also known as homogeneous pairs, an extension of both modules and splits). It is shown how a unique decomposition tree represents the bimodular decomposition of a bipartite graph, with strong analogs with modular decomposition of graphs. An O(mn 3) algorithm for this decomposition is provided. At least a classification of the 2-modules of a bipartite graph is given.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Fouquet, J., Giakoumakis, V., Vanherpe, J.: Bipartite graphs totally decomposable by canonical decomposition. International Journal of Fundations of Computer Science 10, 513–534 (1999)
Cunningham, W., Edmonds, J.: A combinatorial decomposition theory. Canadian Journal of Mathematics 32, 734–765 (1980)
Chein, M., Habib, M., Maurer, M.C.: Partitive hypergraphs. Discrete Mathematics 37, 35–50 (1981)
Mohring, R., Radermacher, F.: Substitution decomposition for discrete structures and connections with combinatorial optimization. Annals of Discrete Mathematics 19, 257–356 (1984)
Cournier, A., Habib, M.: A new linear Algorithm for Modular Decomposition. Lectures notes in Computer Science, pp. 68–84 (1994)
Dahlhaus, E., Gustedt, J., McConnell, R.: Efficient and practical modular decomposition. In: SODA 1997, pp. 26–35 (1997)
McConnell, R., Spinrad, J.: Modular decomposition and transitive orientation. Discrete Mathematics 201, 189–241 (1999)
McConnell, R.M., de Montgolfier, F.: Linear-time modular decomposition of directed graphs. To appear in Discrete Applied Mathematics (2003)
Sumner, D.P.: Graphs indecomposable with respect to the X-join. Discrete Mathematics 6, 281–298 (1973)
de Montgolfier, F.: Décomposition modulaire des graphes. Théorie, extensions et algorithmes. PhD thesis, Université Montpellier II (2003) (in French), available at http://www.lirmm.fr/~montgolfier
Chvátal, V., Sbihi, N.: Bull-free berge graphs are perfect. Graphs Combinatorics 3, 127–139 (1987)
Everett, H., Klein, S., Reed, B.: An algorithm for finding homogeneous pairs. Discrete Applied Mathematics 72, 209–218 (1997)
Cunningham, W.: Decomposition of directed graphs. SIAM Journal of algebraic and discrete methods 3, 214–228 (1982)
Dahlhaus, E.: Parallel algorithms for hierarchical clustering, and applications to split decomposition and parity graph recognition. Journal of Algorithms 36, 205–240 (2000)
Lozin, V.: On maximum induced matchings in bipartite graphs. Information Processing Letters 81, 7–11 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2004 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Fouquet, JL., Habib, M., de Montgolfier, F., Vanherpe, JM. (2004). Bimodular Decomposition of Bipartite Graphs. In: Hromkovič, J., Nagl, M., Westfechtel, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2004. Lecture Notes in Computer Science, vol 3353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-30559-0_10
Download citation
DOI: https://doi.org/10.1007/978-3-540-30559-0_10
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-24132-4
Online ISBN: 978-3-540-30559-0
eBook Packages: Computer ScienceComputer Science (R0)