Abstract
Let G = (V 0, V 1, E) be a connected bipartite graph, where V 0, V 1 is the bipartition of the vertex set V(G) into independent sets. A bipartite drawing of G consists of placing the vertices of V 0 and V 1 into distinct points on two parallel lines x o, x 1, respectively, and then drawing each edge with one straight line segment which connects the points of x 0 and x 1 where the endvertices of the edge were placed. The bipartite crossing number of G, denoted by bcr(G) is the minimum number of crossings of edges over all bipartite drawings of G. We develop a new lower bound method for estimating bcr(G). It relates bipartite crossing numbers to edge isoperimetric inequalities and Laplacian eigenvalues of graphs. We apply the method, which is suitable for “well structured” graphs, to hypercubes and 2-dimensional meshes. E.g. for the n-dimensional hypercube graph we get n4n−2−O(4n) ≤ bcr(Q n) ≤ n4n−1. We also consider a more general setting of the method which uses eigenvalues, but as a trade-off for generality, often gives weaker results.
The research of the first author was supported in part by the NSF grant CCR 9528228.
The research of the 2nd and the 4th author was partially supported by the Alexander von Humboldt Foundation, by the Slovak Scientific Grant Agency grant No. 95/5305/277 and by grant of EU INCO-COP 96-0195.
The research of the third author was supported in part by the NSF grant DMS 9701211 and the Hungarian NSF grants T 016 358 and T 019 367.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Ahlswede, R., Bezrukov, S. L., Edge isoperimetric theorems for integer point arrays, Appl. Math. Lett. 8 (1995), 75–80.
Bezrukov, S. L., Edge isoperimetric problems on graphs, Technical Report, Department of Computer Science, University of Paderborn, 1997.
Bollobás, B., Combinatorics, Chapter 16, Cambridge Uni. Press, 1986.
Bollobás, B., Leader, I., Edge-isoperimetric inequalities in the grid, Combinatorica 11 (1991), 299–314.
Bollobás, B., Leader, I., Matchings and paths in cubes, SIAM J. Discrete Mathematics, to appear.
Brandenburg, F. J., Jünger, M., Mutzel, P., Algorithms for automatic graph drawing, Technical Report, Max Planck Institute, MPI-I-97-1-007, Saarbrücken, March 1997, (in German).
Catarci, T., The assignment heuristics for crossing reduction, IEEE transactions on Systems, Man and Cybernetics 25 (1995), 515–521.
Chung, F. R. K., Spectral Graph Theory, Regional Conference Series in Mathematics Number 92, American Mathematical Society, Providence, RI, 1997.
Chung, F. R. K., Füredi, Z., Graham, R. L., Seymour, P. D., On induced subgraphs of the cube, J. Combinatorial Theory (A) 49 (1988), 180–187.
Di Battista, J., Eades, P., Tamassia, R., Tollis, I.G., Algorithms for drawing graphs: an annotated bibliography, Computational Geometry 4 (1994), 235–282.
Eades, P., Wormald, N., Edge crossings in drawings of bipartite graphs, Algorithmica 11 (1994), 379–403.
Garey, M.R., Johnson, D.S., Crossing number is NP-complete, SIAM J. Algebraic and Discrete Methods 4 (1983), 312–316.
Harary, F., Determinants, permanents and bipartite graphs, Mathematical Magazine 42 (1969), 146–148.
Harary, F., Schwenk, A., A new crossing number for bipartite graphs, Utilitas Mathematica 1 (1972), 203–209.
Harper, L. H., Optimal assignements of numbers to vertices, SIAM J. Applied Mathematics 12 (1964), 131–135.
Jünger, M., Mutzel, P., Exact and heuristic algorithm for 2-layer straightline crossing number, in: Proc. Graph Drawing'95, Lecture Notes in Computer Science 1027, Springer Verlag, Berlin, 1996, 337–348.
Juvan, M., Mohar, B., Optimal linear labelings and eigenvalues of graphs, Discrete Applied Mathematics 36 (1992), 153–168.
Muradyan, D.O., Piliposian, T.E., Minimal numberings of vertices of a rectangular lattice, Akad. Nauk Armjan. SSR Doklady 70 (1980), 21–27, (in Russian).
May, M., Szkatula, K., On the bipartite crossing number, Control and Cybernetics 17 (1988), 85–98.
Shahrokhi, F., Sýkora, 0., Székely, L. A., Vr6, On bipartite crossings, biplanar subgraphs, and the linear arrangement problem, in: Proc. 5th Workshop Algorithms and Data Structures, (WADS'97), August 6–8, 1997 Halifax, Nova Scotia, Canada, Lecture Notes in Computer Science Vol. 1272, Springer-Verlag, 55–68.
F. Shahrokhi, L. A. Székely, I. Vrt'o, Crossing numbers of graphs, lower bound techniques and algorithms: a survey, in: Proc. DIMACS Workshop on Graph Drawing'94, Lecture Notes Computer Science 894, Springer Verlag, Berlin, 1995, 131–142.
Spinrad, J., Brandstädt, A., Stewart, L., Bipartite permutation graphs, Discrete Applied Mathematics 19, 1987, 279–292.
Sugiyama, K., Tagawa, S., Toda, M., Methods for visual understanding of hierarchical systems structures, IEEE Transactions on Systems, Man and Cybernetics 11 (1981), 109–125.
Tutte, W. T., Graph Theory, Addison Wesley Publishing Company, Reading, 1984.
Warfield, J., Crossing theory and hierarchy mapping, IEEE Dnnsactions on Systems, Man and Cybernetics 7 (1977), 502–523.
Watkins, M.E., A special crossing number for bipartite graphs: a research problem, Annals of New York Academy Sciences 175 (1970), 405–410.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1997 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Shahrokhi, F., Sykora, O., Székely, L.A., Vrt'o, I. (1997). Bipartite crossing numbers of meshes and hypercubes. In: DiBattista, G. (eds) Graph Drawing. GD 1997. Lecture Notes in Computer Science, vol 1353. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-63938-1_48
Download citation
DOI: https://doi.org/10.1007/3-540-63938-1_48
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-63938-1
Online ISBN: 978-3-540-69674-2
eBook Packages: Springer Book Archive