## Three-Dimensional 1-Bend Graph Drawings (2004)

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Venue: | Concordia University |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Morin04three-dimensional1-bend,

author = {Pat Morin and David R. Wood},

title = {Three-Dimensional 1-Bend Graph Drawings},

booktitle = {Concordia University},

year = {2004},

pages = {2004}

}

### OpenURL

### Abstract

We consider three-dimensional grid-drawings of graphs with at most one bend per edge. Under the additional requirement that the vertices be collinear, we prove that the minimum volume of such a drawing is Θ(cn), where n is the number of vertices and c is the cutwidth of the graph. We then prove that every graph has a three-dimensional grid-drawing with O(n 3 / log 2 n) volume and one bend per edge. The best previous bound was O(n 3).

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Citation Context ...20]. Only recently have (non-orthogonal) polyline drawings been considered [11, 13]. Table 1 summarises the best known upper bounds on the volume and bends per edge in polyline drawings. Cohen et al. =-=[3]-=- proved that the complete graph Kn (and hence every nvertex graph) has a straight-line drawing with O(n 3 ) volume, and that Ω(n 3 ) volume is necessary. Dyck et al. [13] recently proved that Kn has a... |

351 |
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Citation Context ... by n = |V (G)| and m = |E(G)|. Graph drawing is concerned with the automatic generation of aesthetically pleasing geometric representations of graphs. Graph drawing in the plane is well-studied (see =-=[4, 17]-=-). Motivated by experimental evidence suggesting that displaying a graph in three dimensions is better than in two [21, 22], and applications including information visualisation [21], VLSI circuit des... |

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Citation Context ...a partial order on E(G). A chain (respectively, antichain) in a partial order is a set of pairwise comparable (incomparable) elements. Thus an antichain in � is exactly a cut in σ. Dilworth’s Theorem =-=[8]-=- states that every partial order with no (k + 1)element antichain can be partitioned into k chains. Thus there is a partition of E(G) into chains E1, E2, . . . , Ec, such that each Ei = (ei,1, ei,2, .... |

140 | Evaluating Stereo and Motion Cues for Visualizing Information Nets in Three Dimensions
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Citation Context ...ic representations of graphs. Graph drawing in the plane is well-studied (see [4, 17]). Motivated by experimental evidence suggesting that displaying a graph in three dimensions is better than in two =-=[21, 22]-=-, and applications including information visualisation [21], VLSI circuit design [18], and software engineering [23], there is a growing body of research in three-dimensional graph drawing. A three-di... |

130 | A survey of graph layout problems
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Citation Context ... called the cut in σ at v. The cutwidth of σ is the maximum size of a cut in σ. The cutwidth of G is the minimum cutwidth of a linear order of V (G). Cutwidth is a widely studied graph parameter (see =-=[7]-=-).sMorin & Wood, 1-Bend Graph Drawings, JGAA, 0(0) 0–0 (2005) 2 Table 1: Volume of 3D polyline drawings of n-vertex graphs with m ≥ n edges. graph family bends per edge volume reference arbitrary 0 O(... |

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Citation Context ... by n = |V (G)| and m = |E(G)|. Graph drawing is concerned with the automatic generation of aesthetically pleasing geometric representations of graphs. Graph drawing in the plane is well-studied (see =-=[4, 17]-=-). Motivated by experimental evidence suggesting that displaying a graph in three dimensions is better than in two [21, 22], and applications including information visualisation [21], VLSI circuit des... |

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Citation Context ...ic representations of graphs. Graph drawing in the plane is well-studied (see [4, 17]). Motivated by experimental evidence suggesting that displaying a graph in three dimensions is better than in two =-=[21, 22]-=-, and applications including information visualisation [21], VLSI circuit design [18], and software engineering [23], there is a growing body of research in three-dimensional graph drawing. A three-di... |

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Citation Context ...ce suggesting that displaying a graph in three dimensions is better than in two [21, 22], and applications including information visualisation [21], VLSI circuit design [18], and software engineering =-=[23]-=-, there is a growing body of research in three-dimensional graph drawing. A three-dimensional polyline grid-drawing of a graph, henceforth called a polyline drawing, represents the vertices by distinc... |

44 |
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Citation Context ...otivated by experimental evidence suggesting that displaying a graph in three dimensions is better than in two [21, 22], and applications including information visualisation [21], VLSI circuit design =-=[18]-=-, and software engineering [23], there is a growing body of research in three-dimensional graph drawing. A three-dimensional polyline grid-drawing of a graph, henceforth called a polyline drawing, rep... |

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Citation Context ...f and only if x and y are coprime. Let φ(x) be the number of positive integers less than x that are coprime with x (Euler’s φ function). Thus Nx ≥ φ(x), and N = X� x=1 Nx ≥ X� x=1 φ(x) ≈ 3X2 π 2 (See =-=[15]-=- for a proof that � X x=1 φ(x) ≈ 3X2 /π 2 .) If X ≥ Y/2, then N ≥ 3XY/2π 2 , and we are done. Now assume that Y ≥ 2X. If x and y are coprime, then x and y + x are coprime. Thus Nx ≥ ⌊Y/x⌋ · φ(x). Thus... |

32 | Track layouts of graphs
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Citation Context ...t is not straightforward to compare the volume bound in Theorem 2 with the O(kqm) bound by Dujmović and Wood [11] for k-colourable q-queue graphs (see Table 1). However, since k ≤ 4q and m ≤ 2qn (see =-=[10]-=-), we have that O(kqm) ⊆ O(q 3 n), and thus the O(kqm) bound by Dujmović and Wood [11] is no more than the bound in Theorem 2 whenever the graph has a O((n/ log n) 2/3 )-queue layout. On the other han... |

27 | Layout of graphs with bounded tree-width
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Citation Context ...number 0 O(m 2/3 n) Dujmović and Wood [12] bounded maximum degree 0 O(n 3/2 ) Dujmović and Wood [12] H-minor free (H fixed) 0 O(n 3/2 ) Dujmović and Wood [12] bounded treewidth 0 O(n) Dujmović et al. =-=[9]-=- k-colourable q-queue 1 O(kqm) Dujmović and Wood [11] arbitrary 1 O(nm) Dujmović and Wood [11] cutwidth c 1 O(cn) Theorem 1 arbitrary 1 O(n 3 / log 2 n) Theorem 2 q-queue 2 O(qn) Dujmović and Wood [11... |

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Citation Context ...ings have positive volume. This paper continues the study of upper bounds on the volume and number of bends per edge in polyline drawings. The volume of straight-line drawings has been widely studied =-=[1-5, 8, 9, 11, 12, 15, 17, 19, 20-=-]. Only recently have (non-orthogonal) polyline drawings been considered [10, 14]. Table 1 summarises the best known upper bounds on the volume and bends per edge in polyline drawings. School of Comp... |

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Citation Context ...er edge volume reference arbitrary 0 O(n 3 ) Cohen et al.[3] arbitrary 0 O(m 4/3 n) Dujmović and Wood[11] maximum degree ∆ 0 O(∆mn) Dujmović and Wood[11] bounded chromatic number 0 O(n 2 ) Pach et al.=-=[19]-=- bounded chromatic number 0 O(m 2/3 n) Dujmović and Wood[11] bounded maximum degree 0 O(n 3/2 ) Dujmović and Wood[11] H-minor free (H fixed) 0 O(n 3/2 ) Dujmović and Wood[11] bounded treewidth 0 O(n) ... |

20 | Three-dimensional grid drawings with sub-quadratic volume
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Citation Context ...9 Table 1: Volume of 3D polyline drawings of n-vertex graphs with m ≥ n edges. graph family bends per edge volume reference arbitrary 0 O(n 3 ) Cohen et al.[3] arbitrary 0 O(m 4/3 n) Dujmović and Wood=-=[11]-=- maximum degree ∆ 0 O(∆mn) Dujmović and Wood[11] bounded chromatic number 0 O(n 2 ) Pach et al.[19] bounded chromatic number 0 O(m 2/3 n) Dujmović and Wood[11] bounded maximum degree 0 O(n 3/2 ) Dujmo... |

19 | The maximum number of edges in a three-dimensional grid-drawing
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Citation Context ...13] asked the interesting question: what is the minimum volume in a 1-bend drawing of Kn? The best known upper bound at the time was O(n 3 ), while Ω(n 2 ) is the best known lower bound. (Bose et al. =-=[1]-=- proved that all polyline drawings have Ω(n + m) volume.) In this paper we prove two results. The first concerns collinear polyline drawings in which all the vertices are in a single line. Let G be a ... |

19 | 3D straight-line grid drawing of 4-colorable graphs - CALAMONERI, STERBINI - 1997 |

18 | Tree-partitions of k-trees with applications in graph layout - DUJMOVIĆ, WOOD |

12 | Drawing series-parallel graphs on restricted integer 3D grids - GIACOMO |

11 | Track drawings of graphs with constant queue number - GIACOMO, MEIJER |

11 |
Three-dimensional graph drawing, Algorithmica 17
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Citation Context ...ton.ca. Research supported by NSERC. 1 Table 1: Volume of 3D polyline drawings of graphs with n vertices and m n edges. graph family bends per edge volume reference arbitrary 0 O(n 3 ) Cohen et al. [3] arbitrary 0 O(m 4=3 n) Dujmovic and Wood [12] maximum degree 0 O(mn) Dujmovic and Wood [12] bounded chromatic number 0 O(n 2 ) Pach et al. [19] bounded chromatic number 0 O(m 2=3 n) Dujmovic an... |

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queue and tracks: Layouts of graph subdivisions
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Citation Context ... volume and number of bends per edge in polyline drawings. The volume of straight-line drawings has been widely studied (see [6]). Only recently have (nonorthogonal) polyline drawings been considered =-=[4, 8]-=-. Table 1 summarises the best known upper bounds on the volume and bends per edge in polyline drawings. Cohen et al. [2] proved that the complete graph Kn (and hence every n-vertex graph) has a straig... |

9 |
queues and tracks: Layouts of graph subdivisions
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Citation Context ...ends per edge in polyline drawings. The volume of straight-line drawings has been widely studied [1–3, 6, 9, 11, 12, 14, 16, 19]. Only recently have (non-orthogonal) polyline drawings been considered =-=[4, 12, 13]-=-. Table 1 summarises the best known upper bounds on the volume and bends per edge in polyline drawings. Cohen et al.[3] proved that the complete graph Kn (and hence every nvertex graph) has a straight... |

8 | Drawing Kn in three dimensions with one bend per edge
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Citation Context ...ends per edge in polyline drawings. The volume of straight-line drawings has been widely studied [1–3, 6, 9, 11, 12, 14, 16, 19]. Only recently have (non-orthogonal) polyline drawings been considered =-=[4, 12, 13]-=-. Table 1 summarises the best known upper bounds on the volume and bends per edge in polyline drawings. Cohen et al.[3] proved that the complete graph Kn (and hence every nvertex graph) has a straight... |

7 |
Géza Tóth. Three-dimensional grid drawings of graphs
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Citation Context ...dge volume reference arbitrary 0 O(n 3 ) Cohen et al. [3] arbitrary 0 O(m 4/3 n) Dujmović and Wood [12] maximum degree ∆ 0 O(∆mn) Dujmović and Wood [12] bounded chromatic number 0 O(n 2 ) Pach et al. =-=[20]-=- bounded chromatic number 0 O(m 2/3 n) Dujmović and Wood [12] bounded maximum degree 0 O(n 3/2 ) Dujmović and Wood [12] H-minor free (H fixed) 0 O(n 3/2 ) Dujmović and Wood [12] bounded treewidth 0 O(... |

6 | Drawing Kn in three dimensions with two bends per edge
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Citation Context ...s per edge in polyline drawings. The volume of straight-line drawings has been widely studied [1–3, 5, 6, 9, 11, 12, 14, 16, 20]. Only recently have (non-orthogonal) polyline drawings been considered =-=[11, 13]-=-. Table 1 summarises the best known upper bounds on the volume and bends per edge in polyline drawings. Cohen et al. [3] proved that the complete graph Kn (and hence every nvertex graph) has a straigh... |

5 | Computing straight-line 3D grid drawings of graphs in linear volume. Computational Geometry - Giacomo, Liotta, et al. |

4 |
Layout of graphs with bounded tree-width
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Citation Context ...ings have positive volume. This paper continues the study of upper bounds on the volume and number of bends per edge in polyline drawings. The volume of straight-line drawings has been widely studied =-=[1-5, 8, 9, 11, 12, 15, 17, 19, 20-=-]. Only recently have (non-orthogonal) polyline drawings been considered [10, 14]. Table 1 summarises the best known upper bounds on the volume and bends per edge in polyline drawings. School of Comp... |

2 |
Graph layout problems
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- 1992
(Show Context)
Citation Context ... called the cut in at v. The cutwidth of is the maximum size of a cut in . The cutwidth of G is the minimum cutwidth of a linear order of V (G). Cutwidth is a widely studied graph parameter (see [6]). Theorem 1. Let G be a graph with n vertices and cutwidth c. The minimum volume for a 1-bend collinear drawing of G is (cn). Theorem 1 represents a qualitative improvement over the O(nm) volume bou... |

1 |
queues and tracks: Layouts of graph subdivisions. Discrete Math. Theor. Comput. Sci., to appear. Layouts of graph subdivisions
- Stacks
- 2004
(Show Context)
Citation Context ...s per edge in polyline drawings. The volume of straight-line drawings has been widely studied [1–3, 5, 6, 9, 11, 12, 14, 16, 20]. Only recently have (non-orthogonal) polyline drawings been considered =-=[11, 13]-=-. Table 1 summarises the best known upper bounds on the volume and bends per edge in polyline drawings. Cohen et al. [3] proved that the complete graph Kn (and hence every nvertex graph) has a straigh... |