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Lower Bounds for the Number of Bends in ThreeDimensional Orthogonal Graph Drawings
, 2003
"... This paper presents the first nontrivial lower bounds for the total number of bends in 3D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3D orthogonal drawings of complete simple graphs and multigraphs, which are tigh ..."
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This paper presents the first nontrivial lower bounds for the total number of bends in 3D orthogonal graph drawings with vertices represented by points. In particular, we prove lower bounds for the number of bends in 3D orthogonal drawings of complete simple graphs and multigraphs, which are tight in most cases. These result are used as the basis for the construction of infinite classes of cconnected simple graphs, multigraphs, and pseudographs (2 ≤ c ≤ 6) of maximum degree Δ (3 ≤ Δ ≤ 6), with lower bounds on the total number of bends for all members of the class. We also present lower bounds for the number of bends in general position 3D orthogonal graph drawings. These results have significant ramifications for the `2bends problem', which is one of the most important open problems in the field.
Orthogonal drawings with few layers
 PROC. 9TH INTERNATIONAL SYMP. ON GRAPH DRAWING (GD '01
, 2002
"... In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very smal ..."
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In this paper, we study 3dimensional orthogonal graph drawings. Motivated by the fact that only a limited number of layers is possible in VLSI technology, and also noting that a small number of layers is easier to parse for humans, we study drawings where one dimension is restricted to be very small. We give algorithms to obtain pointdrawings with 3layers and 4 bends per edge, and algorithms to obtain boxdrawings with 2 layers and 2 bends per edge. Several other related results are included as well. Our constructions have optimal volume, which we prove by providing lower bounds.
The Topology of Bendless ThreeDimensional Orthogonal Graph Drawing
, 2007
"... We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; ..."
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We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; we call a graph with such an embedding an xyz graph. We describe a correspondence between xyz graphs and facecolored embeddings of the graph onto twodimensional manifolds, and we relate bipartiteness of the xyz graph to orientability of the underlying topological surface. Using this correspondence, we show that planar graphs are xyz graphs if and only if they are bipartite, cubic, and threeconnected, and that it is NPcomplete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n2 n/2) for testing whether a given graph is an xyz graph.
Drawing a graph in a hypercube
, 2004
"... A ddimensional hypercube drawing of a graph represents the vertices by distinct points in {0, 1} d, such that the linesegments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to ..."
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A ddimensional hypercube drawing of a graph represents the vertices by distinct points in {0, 1} d, such that the linesegments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections. 1
Imbalance is Fixed Parameter Tractable
"... Abstract. In the Imbalance Minimization problem we are given a graph G = (V, E) and an integer b and asked whether there is an ordering v1... vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the ..."
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Abstract. In the Imbalance Minimization problem we are given a graph G = (V, E) and an integer b and asked whether there is an ordering v1... vn of V such that the sum of the imbalance of all the vertices is at most b. The imbalance of a vertex vi is the absolute value of the difference between the number of neighbors to the left and right of vi. The problem is also known as the Balanced Vertex Ordering problem and it finds many applications in graph drawing. We show that this problem is fixed parameter tractable and provide an algorithm that runs in time 2 O(b log b) · n O(1). This resolves an open problem of Kára et al. [COCOON 2005]. 1
Bibliography
"... Automatic clustering of languages. Computational Linguistics, 18(3):339352. Berge, C. (1993). Graphs. North Holland, Amsterdam, 3rd edition. Berger, B., and Shor, P. (1990). Approximation algorithms for the maximum acyclic subgraph problem. In Proceedings of the 1st ACMSIAM Symposium on Discrete ..."
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Automatic clustering of languages. Computational Linguistics, 18(3):339352. Berge, C. (1993). Graphs. North Holland, Amsterdam, 3rd edition. Berger, B., and Shor, P. (1990). Approximation algorithms for the maximum acyclic subgraph problem. In Proceedings of the 1st ACMSIAM Symposium on Discrete Algorithms (SODA'90), pages 236243. Bertolazzi, P., Cohen, R. F., Di Battista, G., Tamassia, R., and Tollis, I. G. (1994a). How to draw a seriesparallel digraph. International Journal of Computational Geometry and Applications, 4:385402. Bertolazzi, P., Di Battista, G., and Didimo, W. (1997). Computing orthogonal drawings with the minimum number of bends. In Proceedings of the 5th Workshop on Algorithms and Data Structures (WADS'97), Spinger LNCS 1272, pages 331344. Bertolazzi, P., Di Battista, G., Liotta, G., and Mannino, C. (1994b). Upward drawings of triconnected digraphs. Algorithmica, 6(12):476
BoundedDegree Book Embedings and ThreeDimensional Orthogonal Graph Drawing
 Proc. 9th International Symp. on Graph Drawing (GD '01), volume 2265 of Lecture Notes in Comput. Sci
, 2002
"... A book embedding of a graph consists of a linear orderin of the vertices along a line in 3space (the spine), and an assignment of edges to halfplanes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph $G=(V,E)$ ..."
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A book embedding of a graph consists of a linear orderin of the vertices along a line in 3space (the spine), and an assignment of edges to halfplanes with the spine as boundary (the pages), so that edges assigned to the same page can be drawn on that page without crossings. Given a graph $G=(V,E)$, let $f:V\rightarrow\mathbb{N}$ be a function such that $1\leq f(v)\leq\deg(v)$. We present a Las Vegas algorithm which produces a book embedding of $G$ with \Oh{\sqrt{E\cdot\max_v\ceil{\deg(v)/f(v)}}} pages, such that at most $f(v)$ edges incident to a vertex $v$ are on a single page. This algorithm generalises existing results for book embeddings. We apply this algorithm to produce 3D orthogonal drawings with one bend per edge and \Oh{V^{3/2}E} volume, and \emph{singlerow} drawings with two bends per edge and the same volume. In the produced drawings each edge is entirely contained in some $Z$plane; such drawings are without socalled \emph{crosscuts}, and are particularly appropriate for applications in multilayer VLSI. Using a different approach, we achieve two bends per edge with \Oh{VE} volume but with crosscuts. These results establish improved bounds for the volume of 3D orthogonal graph drawings.
Fully Dynamic 3Dimensional . . .
 J. GRAPH ALGORITHMS APPL
, 2000
"... In a 3dimensional orthogonal drawing of a graph, vertices are mapped to grid points on a 3dimensional rectangular integer lattice and edges are routed along integer grid lines. In this paper, we present a technique that produces a 3D orthogonal drawing of any graph with n vertices of degree 6 or l ..."
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In a 3dimensional orthogonal drawing of a graph, vertices are mapped to grid points on a 3dimensional rectangular integer lattice and edges are routed along integer grid lines. In this paper, we present a technique that produces a 3D orthogonal drawing of any graph with n vertices of degree 6 or less, using at most 6 bends per edge route and in a volume bounded by O(n 2 ). The advantage of our strategy over previous drawing methods is that our method is fully dynamic, allowing both insertion and deletion of vertices and edges, while maintaining the volume and bend bounds. The drawing can be obtained in O(n) time and insertions/deletions are performed in O(1) time. Multiple edges and self loops are permitted. Three related constructions are also presented: a more elaborate construction that uses only 5 bends per edge, a simpler, more balanced drawing that requires at most 7 bends per edge, and a technique for displaying directed graphs.