Three-dimensional orthogonal graph drawing algorithms

Eades, Peter; Symvonis, Antonios; Whitesides, Sue

doi:10.1016/s0166-218x(00)00172-4

Cited by 44 publications

(76 citation statements)

References 20 publications

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“…The decomposition into cycle covers follows from a classical result [15] stating that "a regular multigraph G of degree 2k has k edge-disjoint factors", where a factor is a spanning subgraph consisting of vertex-disjoint cycles (see also [2, pp.227]). A constructive proof of the following theorem was given in [8]. Let u be a vertex placed at a grid point.…”

Section: Rac-drawings Of Bounded-degree Graphsmentioning

confidence: 99%

“…4(a) shows a regular directed multigraph G of indegree and outdegree 3 and its cycle covers (C 1 : thin, C 2 : thick, C 3 : dashed). C 1 consists of cycles c 1 : (5, 1, 2, 3, 4, 5) and c 2 : (6,7,8,9,6). We set 4 as the topmost vertex of c 1 since edge (4, 6) of C 2 has vertex 6 of c 2 as its destination.…”

Section: Theorem 5 Every N-vertex Graph With Degree At Most 6 Admitsmentioning

confidence: 99%

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On the Perspectives Opened by Right Angle Crossing Drawings

et al. 2010

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Abstract. Right Angle Crossing (RAC) drawings are polyline drawings where each crossing forms four right angles. RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of the drawing if the edges cross at right angles. We investigate to what extent RAC drawings can help in overcoming the limitations of widely adopted planar graph drawing conventions, providing both positive and negative results. First, we prove that there exist acyclic planar digraphs not admitting any straight-line upward RAC drawing and that the corresponding decision problem is NP-hard. Also, we show digraphs whose straightline upward RAC drawings require exponential area. Second, we study if RAC drawings allow us to draw bounded-degree graphs with lower curve complexity than the one required by more constrained drawing conventions. We prove that every graph with vertex-degree at most 6 (at most 3) admits a RAC drawing with curve complexity 2 (resp. 1) and with quadratic area. Third, we consider a natural non-planar generalization of planar embedded graphs. Here we give bounds for curve complexity and area different from the ones known for planar embeddings.

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Section: Rac-drawings Of Bounded-degree Graphsmentioning

confidence: 99%

Section: Theorem 5 Every N-vertex Graph With Degree At Most 6 Admitsmentioning

confidence: 99%

On the Perspectives Opened by Right Angle Crossing Drawings

et al. 2010

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“…1. Orthogonal point-drawings have been studied extensively in two dimensions (see [7]) and to a lesser extent in three dimensions [10,11,18,21]. However, Ddimensional point-drawings can only exist for graphs with maximum degree at most 2D.…”

Section: Introductionmentioning

confidence: 99%

“…The trade-off between the maximum number of bends per edge route and the bounding box volume apparent in 3-D point-drawing algorithms [10,11], is seen to a lesser extent in 3-D orthogonal box-drawings. Biedl et al [3] Papakostas and Tollis [18] first established that every graph has a degreerestricted 3-D orthogonal drawing.…”

Section: Introductionmentioning

confidence: 99%

“…In [23] an algorithm which generalizes the COMPACT point-drawing algorithm of Eades et al [10,11] produces 6-bend 3-D orthogonal line-drawings of multigraphs with bounding box volume O(m 2 / √ n), which is the best known bounding box volume upper bound for graphs with O(n 7/4 ) edges. Our layout-based algorithm improves the best known bound for the degreerestriction of vertices in bounded aspect ratio 3-D orthogonal drawings.…”

Section: Introductionmentioning

confidence: 99%

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Multi-dimensional Orthogonal Graph Drawing with Small Boxes

Wood

1999

Graph Drawing

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Abstract. In this paper we investigate the general position model for the drawing of arbitrary degree graphs in the D-dimensional (D ≥ 2) orthogonal grid. In this model no two vertices lie in the same grid hyperplane. We show that for D ≥ 3, given an arbitrary layout and initial edge routing a crossing-free orthogonal drawing can be determined. We distinguish two types of algorithms. Our layout-based algorithm, given an arbitrary fixed layout, determines a degree-restricted orthogonal drawing with each vertex having aspect ratio two. Using a balanced layout this algorithm establishes improved bounds on the size of vertices for 2-D and 3-D drawings. Our routing-based algorithm produces 2-degreerestricted 3-D orthogonal drawings. One advantage of our approach in 3-D is that edges are typically routed on each face of a vertex; hence the produced drawings are more truly three-dimensional than those produced by some existing algorithms.

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Complexity Results for Three-Dimensional Orthogonal Graph Drawing

Patrignani

2006

Graph Drawing

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We introduce the 3SAT reduction framework which can be used to prove the NP-hardness of finding three-dimensional orthogonal drawings with specific constraints. We use it to show that finding a drawing of a graph whose edges have a fixed shape is NP-hard. Also, it is NP-hard finding a drawing of a graph with nodes at prescribed positions when a maximum of two bends per edge is allowed. We comment on the impact of these results on the two open problems of determining whether a graph always admits a 3D orthogonal drawing with at most two bends per edge and of characterizing orthogonal shapes admitting a drawing without intersections.

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Three-dimensional orthogonal graph drawing algorithms

Cited by 44 publications

References 20 publications

On the Perspectives Opened by Right Angle Crossing Drawings

On the Perspectives Opened by Right Angle Crossing Drawings

Multi-dimensional Orthogonal Graph Drawing with Small Boxes

Complexity Results for Three-Dimensional Orthogonal Graph Drawing

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