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Drawing graphs in two layers. (English) Zbl 0819.68086
Summary: Let \(G= (U, L, E)\) be a bipartite graph with vertex set \(U\cup L\) and edge set \(E\subseteq U\times L\). A typical convention for drawing \(G\) is to put the vertices of \(U\) on the line and the vertices of \(L\) on a separate, parallel line and then to represent edges by placing open straight line segments between the vertices that determine them. In this convention, a drawing is biplanar if edges do not cross, and a subgraph of \(G\) is biplanar if it has a biplanar drawing. The main results of this paper are the following: (1) it is NP-complete to determine whether \(G\) has a biplanar subgraph with at least \(K\) edges; (2) it is also NP- complete to determine whether \(G\) has such a subgraph when the position for the vertices in either \(U\) or \(L\) are specified; (3) when the position of the vertices in both \(U\) and \(L\) are specified, the problem can be solved in polynomial time by transformation to the longest ascending subsequence problem.

MSC:
68R10 Graph theory (including graph drawing) in computer science
68Q25 Analysis of algorithms and problem complexity
05C10 Planar graphs; geometric and topological aspects of graph theory
Keywords:
bipartite graph
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[1] Angluin, D.; Valiant, L.G., Fast probabilistic algorithms for Hamilton circuits and matchings, J. comput. system sci., 18, 155-193, (1979) · Zbl 0437.05040
[2] Atallah, M.J.; Kosaraju, S.R., An efficient algorithm for maxdominance with applications, Algorithmica, 4, 221-236, (1989) · Zbl 0664.68070
[3] Dijkstra, E.W., Some beautiful arguments using mathematical induction, Acta inform., 13, 1-8, (1980) · Zbl 0435.68055
[4] Eades, P.; Kelly, D., Heuristics for reducing crossings in 2-layered networks, Ars combin., 21A, 89-98, (1986) · Zbl 0598.05032
[5] Eades, P.; Lin, X., How to draw a directed graph, (), 13-17
[6] Eades, P.; Mckay, B.D.; Wormald, N.C., On an edge crossing problem, (), 327-334
[7] Eades, P.; Sugiyama, K., How to draw a directed graph, J. inform. process., 13, 424-437, (1990) · Zbl 0764.68114
[8] Eades, P.; Tamassia, R., Algorithms for drawing graphs: an annotated bibliography, () · Zbl 0804.68001
[9] Eades, P.; Wormald, N.C., Edge crossings in drawings of bipartite graphs, () · Zbl 0804.68107
[10] Gansner, E.R.; North, S.C.; Vo, K.P., DAG — a program that draws directed graphs, Software practice experience, 18, 1047-1062, (1988) · Zbl 0661.68067
[11] Garey, M.R.; Johnson, D.S., Computers and intractability: A guide to the theory of NP-completeness, (1979), Freeman New York · Zbl 0411.68039
[12] Garey, M.R.; Johnson, D.S., Crossing number is NP-complete, SIAM J. algebraic discrete methods, 4, 312-316, (1983) · Zbl 0536.05016
[13] Gschwind, D.J.; Murtagh, T.P., A recursive algorithm for drawing hierarchical directed graphs, ()
[14] Karger, D.; Motwani, R.; Ramkumar, G.D.S., On approximating the longest path in a graph, (), 421-432 · Zbl 0876.68083
[15] Kelly, D., A view to graph layout problems, ()
[16] Lou, R.D.; Sarrafzadeh, M.; Lee, D.T., An optimal algorithm for the maximum two-chain problem, SODA 90, proc. 1st ann. ACM-SIAM symp. on discrete algorithms, 149-158, (1990), San Francisco · Zbl 0800.68473
[17] Makinen, E., Experiments in drawing 2-level hierarchical graphs, () · Zbl 0723.68083
[18] Messinger, E., Automatic layout of large directed graphs, ()
[19] Orlowski, M.; Pachter, M., An algorithm for the determination of a longest increasing subsequence in a sequence, Comput. math. appl., 17, 1073-1075, (1989) · Zbl 0671.05002
[20] Paulich, F.N.; Tichy, W.F., EDGE: an extendible directed graph editor, Software practice experience, 20, 63-88, (1990)
[21] Rowe, L.A.; Davis, M.; Messinger, E.; Meyer, C.; Spirakis, C.; Tuan, A., A browser for directed graphs, Software practice experience, 17, 61-76, (1987)
[22] Sechen, C., VLSI placement and global routing using simulated annealing, (1988), Kluwer Dordrecht
[23] Sugiyama, K., A cognitive approach for graph drawing, Cybernet. systems, 18, 447-488, (1987)
[24] Sugiyama, K.; Misue, K., Visualizing structural information: hierarchical drawing of a compound digraph, (), IIAS-SIS
[25] Sugiyama, K.; Tagawa, S.; Toda, M., Methods for visual understanding of hierarchical system structures, IEEE trans. systems man cybernet, SMC-11, 109-125, (1981)
[26] Ullman, J.D., Computational aspects of VLSI, (1984), Computer Science Press Rockville, MD · Zbl 0539.68021
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