## Track drawings of graphs with constant queue number

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Venue: | In [13 |

Citations: | 10 - 0 self |

### BibTeX

@INPROCEEDINGS{Giacomo_trackdrawings,

author = {Emilio Di Giacomo},

title = {Track drawings of graphs with constant queue number},

booktitle = {In [13},

year = {},

pages = {214--225}

}

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### Abstract

### Citations

515 | A heuristic for graph drawing
- Eades
- 1984
(Show Context)
Citation Context ... body of combinatorial results, structures, algorithmic techniques, and software systems, the research on 3D drawings can still be considered in its early stages [5, 16]. Cohen, Eades, Lin and Ruskey =-=[3]-=- showed that every graph admits crossing-free 3D drawing on an integer grid of O ¡ n 3¢ volume, and proved that this is asymptotically optimal. The volume of a drawing is measured as the number of gri... |

43 |
Laying out graphs using queues
- HEATH, ROSENBERG
- 1992
(Show Context)
Citation Context ...o compute such a drawing. Wood [18] shows a relationship between tn ¡ G ¢ and another well-studied graph parameter, the queue number qn ¡ G ¢ (i.e. the minimum number of queues in a queue layout of G =-=[15]-=-). He proves that every graph from a proper minor closed family has constant track number if and only if it has constant queue number. By the result of Wood all families of graphs whose queue number i... |

37 | Drawing Graphs - Kaufmann, Wagner - 2001 |

33 |
Comparing queues and stacks as mechanisms for laying out graphs
- HEATH, LEIGHTON, et al.
- 1992
(Show Context)
Citation Context ...e1. Thus it is sufficient to prove that the equation has no solution in order to prove that e0 and e1 do not cross each other. We call this equation co-planarity equation of e0 and e1. A queue layout =-=[14, 15]-=- of a graph G consists of a linear ordering λ of the vertices of G, and a partition of the edges of G into queues, such that no two edges in the same queue are nested with respect to λ. In other words... |

23 | Path-width and threedimensional straight-line grid drawings of graphs
- Dujmović, Morin, et al.
- 2002
(Show Context)
Citation Context ... may not be integer. Chrobak, Goodrich, and Tamassia [2] gave an algorithm for constructing 3D convex drawings of triconnected planar graphs with O(n) volume and non-integer coordinates.Recent papers =-=[6, 7, 8, 9, 10, 18]-=- have considered the following problem: given a grid f such that f is a proper subset of the integer 3D space, which graphs admit a straight line crossing-free drawing with vertices located at thegrid... |

20 | Three Dimensional Grid Drawings of Graphs
- Pach, Thiele, et al.
- 1999
(Show Context)
Citation Context ...urable graphs can be drawn in a 3D grid of O(n2) volume with O(n) aspect ratio and proved alower bound of \Omegas(n1 :5 ) on the volume of such graphs. For r-colourable graphs, Pach, Thiele and T'oth =-=[17]-=- showed abound of q (n2) on the volume. Garg, Tamassia, and Vocca [12] showed that all 4-colorable graphs (and hence all planargraphs) can be drawn in O (n1 :5 ) volume and with O(1) aspect ratio but ... |

16 | Drawing series-parallel graphs on a box
- Giacomo, Liotta, et al.
(Show Context)
Citation Context ... not be integer. Chrobak, Goodrich, and Tamassia [2] gave an algorithm for constructing 3D convex drawings of triconnected planar graphs with O ¡ n ¢ volume and non-integer coordinates. Recent papers =-=[6, 7, 8, 9, 10, 18]-=- have considered the following problem: given a grid φ such that φ is a proper subset of the integer 3D space, which graphs admit a straight line crossing-free drawing with vertices located at the gri... |

15 | Tree-partitions of k-trees with applications in graph layout - Dujmović, Wood - 2003 |

12 | Queue layouts, tree-width, and three-dimensional graph drawing
- Wood
(Show Context)
Citation Context ...ng is a crossing-free 3D straight-line drawing of a graph G on a set of k parallel lines called tracks. The minimum value of k for which G admits a k-track drawing is called the track number of G. In =-=[18]-=- it is proved that every graph from a proper minor closed family has constant track number if and only if it has constant queue number. In this paper we study the track number of well-known families o... |

11 |
Drawing with colors
- Garg, Tamassia, et al.
- 1996
(Show Context)
Citation Context ...ect ratio and proved a lower bound of Ω ¡ n 1£ 5¢ on the volume of such graphs. For r-colourable graphs, Pach, Thiele and Tóth [17] showed a bound of θ ¡ n 2¢ on the volume. Garg, Tamassia, and Vocca =-=[12]-=- showed that all 4-colorable graphs (and hence all planar graphs) can be drawn in O ¡ n 1£ 5¢ volume and with O ¡ 1 ¢ aspect ratio but by using a grid model where the coordinates of the vertices may n... |

8 |
Drawing 2-, 3- and 4-colorable graphs in o(n 2 ) volume
- Calamoneri, Sterbini
- 1997
(Show Context)
Citation Context ...d proved that this is asymptotically optimal. The volume of a drawing is measured as the number of grid-points contained in the smallest axis-aligned box bounding the drawing. Calamoneri and Sterbini =-=[1]-=- showed that all 2-, 3-, and 4-colourable graphs can be drawn in a 3D grid of O ¡ n 2¢ volume with O ¡ n ¢ aspect ratio and proved a lower bound of Ω ¡ n 1£ 5¢ on the volume of such graphs. For r-colo... |

7 | Drawing graphs on two and three lines
- Cornelsen, Schank, et al.
(Show Context)
Citation Context ... 3sv1 v2 v3 v4 (a) v1 v2 v5 v3 v7 (c) v4 v7 v8 v 5 v 6 v8 v6 v1 v3 v6 v4 v7 v5 (b) v1 v2 v5 Figure 1: (a) A graph G. (b) A track assignment of G. (c) A track layout of G. (d) A track drawing of G. In =-=[4]-=- the graphs having track number 2 are characterized, and it is proved that they are a subclass of the outerplanar graphs. The following lemma is an immediate corollary of the result in [4]. Lemma 1 [4... |

7 | Stack and queue layouts of Halin graphs
- GANLEY
- 1995
(Show Context)
Citation Context ...which C is the boundary of the external face. � T is called the characteristic tree of G and C is called the adjoint cycle of G. Figure 4 shows a Halin graph. It is known from the existing literature =-=[11]-=- that 3 queues are always sufficient for a queue layout of a Halin graph and that 2 queues are sometimes necessary. A lower bound on the track number of Halin graphs is 3, since Halin v0 v14 v1 v2 v3 ... |

6 | Convex drawings of graphs in two and three dimensions (preliminary version
- Chrobak, Goodrich, et al.
- 1996
(Show Context)
Citation Context ...hence all planargraphs) can be drawn in O (n1 :5 ) volume and with O(1) aspect ratio but by using a grid model where the coordinates ofthe vertices may not be integer. Chrobak, Goodrich, and Tamassia =-=[2]-=- gave an algorithm for constructing 3D convex drawings of triconnected planar graphs with O(n) volume and non-integer coordinates.Recent papers [6, 7, 8, 9, 10, 18] have considered the following probl... |

4 |
Straight line drawings on restricted integer grids in two and three dimensions
- Felsner, Liotta, et al.
- 2001
(Show Context)
Citation Context ... not be integer. Chrobak, Goodrich, and Tamassia [2] gave an algorithm for constructing 3D convex drawings of triconnected planar graphs with O ¡ n ¢ volume and non-integer coordinates. Recent papers =-=[6, 7, 8, 9, 10, 18]-=- have considered the following problem: given a grid φ such that φ is a proper subset of the integer 3D space, which graphs admit a straight line crossing-free drawing with vertices located at the gri... |

3 | Drawing 2-, 3- and 4-colorable graphs in o(n2) volume
- Calamoneri, Sterbini
- 1997
(Show Context)
Citation Context ...nd proved that this is asymptotically optimal. The volume of a drawing is measured as the numberof grid-points contained in the smallest axis-aligned box bounding the drawing. Calamoneri and Sterbini =-=[1]-=- showed that all 2-, 3-, and 4-colourable graphs can be drawn in a 3D grid of O(n2) volume with O(n) aspect ratio and proved alower bound of \Omegas(n1 :5 ) on the volume of such graphs. For r-coloura... |

2 |
Studies in minimally connected graphs
- Halin
- 1971
(Show Context)
Citation Context ...algorithm consists of a BFS visit and therefore thetime complexity is O (n). \Xis8s5 Halin Graphs In this section we study the track number of a well-investigated family of graphs called Halin Graphs =-=[13]-=-. A Halingraph is a graph such that: ffl every vertex of G has degree greater or equal to 3; ffl G can be decomposed into a spanning tree T of G and a cycle C through the leaves of T ; ffl G has a pla... |

1 |
Graph Drawing. Prentice Hall, Upper Saddle River
- Battista, Eades, et al.
- 1999
(Show Context)
Citation Context ...ast two decades and has produced a rich body of combinatorial results, structures,algorithmic techniques, and software systems, the research on 3D drawings can still be considered in its early stages =-=[5, 16]-=-.Cohen, Eades, Lin and Ruskey [3] showed that every graph admits crossing-free 3D drawing on an integer grid of O(n3) volume, and proved that this is asymptotically optimal. The volume of a drawing is... |