M. B. Dillencourt, D. Eppstein, and D. S. Hirschberg, Geometric thickness of complete graphs. J. Graph Algorithms Appl., 4(3):5--17, 2000. Home/Search Document Details and Download
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On Simultaneous Planar Graph Embeddings - Brass, Cenek, Duncan, Efrat.. (Correct)
....edges drawn in the same layer intersecting only at a common vertex and vertices placed in the same location in all layers. A related graph property is geometric thickness, defined to be the minimum number of layers for which a drawing of G exists having all edges drawn as straight line segments [11]. Finally, the book thickness of a graph G is the minimum number of layers for which a drawing of G exists, in which edges are drawn as straight line segments and vertices are in convex position [2] It has been shown that the book thickness of planar graphs is no greater than four [21] As ....
M. B. Dillencourt, D. Eppstein, and D. S. Hirschberg. Geometric thickness of complete graphs. Journal of Graph Algorithms and Applications, 4(3):5--17, 2000.
Geometric Thickness in a Grid - Wood (2001) (Correct)
....G can be drawn in the plane with edges as straight line segments, and with each edge assigned to a layer so that no two edges in the same layer cross. Geometric thickness was rst introduced under the name of real linear thickness by Kainen [17] and has recently been studied by Dillencourt et al. [11]. Applications of geometric thickness include multilayer VLSI [1, 2] and graph visualization where layers correspond to colours in a drawing. Geometric thickness is closely related to the (graph theoretic) thickness of a graph, de ned to be the minimum number of subgraphs in a partition of the ....
....1 are precisely the planar graphs. Graphs with geometric thickness 2, called doubly linear graphs, are studied in [5, 16] Hutchinson et al. 16] prove that every doubly linear graph has at most 6n 18 edges, and present doubly linear graphs with 6n 20 edges for all n 8. Dillencourt et al. [11] establish lower and upper bounds for the geometric thickness of complete and complete bipartite graphs. It is shown that d n 1 5:646 e (Kn ) d 4 e. Note that their construction has O(n ) area under the vertex resolution rule [D. Eppstein, personal communication] Since (Kn ) 6 ....
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M. B. Dillencourt, D. Eppstein, and D. S. Hirschberg, Geometric thickness of complete graphs. J. Graph Algorithms Appl., 4(3):5-17, 2000.
Geometric Thickness in a Grid of Linear Area - Wood (2001) (Correct)
....segments, and with each edge assigned to a layer so that no two edges in the same layer cross. Geometric thickness was rst introduced under the name of real linear thickness [Kai73] Lower and upper bounds are established for the geometric thickness of complete and complete bipartite graphs in [DEH00]. In particular, it is shown that (Kn ) Kn ) for large n. On the other hand, K a;b ) K a;b ) when a b. Applications of geometric thickness include multilayer VLSI [AKS91] and graph visualisation [DETT99] Another parameter closely related to geometric thickness is that of book ....
....VLSI and visualisation. To measure the area of a drawing we assume a vertex resolution rule; that is, pairs of vertices are at least unit distance apart. A drawing obtained from a book embedding by positioning the vertices around a circle, as discussed above, has O(n ) area. The construction in [DEH00] demonstrating that (Kn ) d 4 e has O(n 6 ) area [D. Eppstein, personal communication ] We prove the following 2 dimensional generalisation of the above mentioned result in [Mal94b] for producing book embeddings. Theorem 1. The vertices of G can be positioned in a 2 n 2 n grid, ....
M. B. Dillencourt, D. Eppstein, and D. S. Hirschberg. Geometric thickness of complete graphs. J. Graph Algorithms Appl., 4(3):5-17, 2000.
Planarizing Graphs - A Survey and Annotated Bibliography - Liebers (2001) (6 citations) (Correct)
....linear . Clearly any doubly linear graph has thickness at most two. Hutchinson et al. HSV96, HSV99] study doubly linear graphs. They show that a doubly linear graph with n vertices has at most 6n 18 edges. Other variations of thickness are discussed in [Hob69, H S82, Hor83, Wes83a, PCK89, DEH00] for example. 6 Crossing Number In graph drawing, but also in other application areas such as VLSI layout, we are interested in a drawing of a given graph with as few edge crossings as possible. Here, we do not allow drawings of graphs where a point in the plane belongs to more than two curves ....
Michael B. Dillencourt, David Eppstein, and Daniel S. Hirschberg. Geometric Thickness of Complete Graphs. Journal of Graph Algorithms and Applications, 4:5--17, 2000.
Really Straight Graph Drawings - Dujmovic, Suderman, Wood (2004) (Correct)
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M. B. Dillencourt, D. Eppstein, and D. S. Hirschberg, Geometric thickness of complete graphs. J. Graph Algorithms Appl., 4(3):5--17, 2000.
Stacks, Queues and Tracks: Layouts of Graph Subdivisions - Dujmovic, Wood (2004) (Correct)
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M. B. DILLENCOURT, D. EPPSTEIN, AND D. S. HIRSCHBERG, Geometric thickness of complete graphs. J. Graph Algorithms Appl., 4(3):5--17, 2000.
Partitions of Complete Geometric Graphs into Plane Trees - Prosenjit Bose Ferran (Correct)
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M. B. DILLENCOURT, D. EPPSTEIN, AND D. S. HIRSCHBERG, Geometric thickness of complete graphs. J. Graph Algorithms Appl., 4(3):5--17, 2000.
Track Layouts of Graphs - Dujmovic, Wood (2004) (Correct)
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M. B. DILLENCOURT, D. EPPSTEIN, AND D. S. HIRSCHBERG, Geometric thickness of complete graphs. J. Graph Algorithms Appl., 4(3):5--17, 2000.
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