## Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs (1999)

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### Other Repositories/Bibliography

Venue: | Algorithmica |

Citations: | 60 - 12 self |

### BibTeX

@INPROCEEDINGS{Eades99straight-linedrawing,

author = {Peter Eades and Qingwen Feng and Xuemin Lin and Hiroshi Nagamochi},

title = {Straight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs},

booktitle = {Algorithmica},

year = {1999},

pages = {113--128},

publisher = {Springer-Verlag}

}

### Years of Citing Articles

### OpenURL

### Abstract

Hierarchical graphs and clustered graphs are useful non-classical graph models for structured relational information. Hierarchical graphs are graphs with layering structures; clustered graphs are graphs with recursive clustering structures. Both have applications in CASE tools, software visualization, and VLSI design. Drawing algorithms for hierarchical graphs have been well investigated. However, the problem of straight-line representation has not been solved completely. In this paper, we answer the question: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? We present an algorithm that constructs such drawings in linear time. Also, we answer a basic question for clustered graphs, that is, does every planar clustered graph admit a planar straight-line drawing with clusters drawn as convex polygons? We provide a method for such drawings based on our algorithm for hierarchical graphs.

### Citations

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Citation Context ...et E of edges, that is, pairs of vertices. Graphs are commonly used to model relations in computing, and many systems for manipulating graphs have recently been developed. Examples include CASE tools =-=[55]-=- knowledge representation systems [31], software visualization tools [54] and VLSI design systems [26]. A graph drawing algorithm reads as input a combinatorial description of a graph, and produces as... |

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Citation Context ...ated by Chibas4 P. Eades, Q. Feng, X. Lin, and H. Nagamochi et al. [8], [9]. More recently, efficient algorithms for planar straight-line drawings were developed by de Fraysseix et al. [23], Schnyder =-=[46]-=-, Chrobak [10] and Kant [32]. These recent methods show that every planar graph admits a straight-line drawing in which each vertex is located at an integer grid point and the whole drawing uses O(n 2... |

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Citation Context ...olygons. Algorithms for such drawings have also been investigated by Chiba et al. [6, 7]. More recently, 3 efficient algorithms for planar straight-line drawings were developed by de Fraysseix et al. =-=[22]-=-, Schnyder [41], Chrobak [8] and Kant [29]. These recent methods show that every planar graph admits a straight-line drawing in which each vertex is located at an integer grid point and the whole draw... |

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Citation Context ...t admits a planar drawing. The planarity property has been the object of much of Graph Theory. For visualization purposes, it is well established that edge crossings significantly inhibit readability =-=[40]-=-, and many algorithms for constructing planar drawings have been developed [2], [33].sStraight-Line Drawing Algorithms for Hierarchical Graphs and Clustered Graphs 3 ANU UNcstle UNSW Universities UTS ... |

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Citation Context ...rossings. One of the basic graph drawing convention consists of representing edges as straight-line segments. For classical graphs, it has been shown independently by Fary [7], Stein [17], and Wagner =-=[21]-=- that every planar graph admits a straight-line drawing without edge crossings. Tutte [20] proved that every 3-connected planar graph admits a planar straight-line drawing where all the face boundarie... |

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Citation Context ...it [44], GraphEd [28] and Diagram Server [3] contain a module for creating straight-line drawings of classical graphs. Intuitively, the eye can follow a straight-line easily; Sugiyama [48] and Batini =-=[51]-=- list straightness of lines as an important aim for graph drawing algorithms. This intuition has been confirmed by human experiments [41], [40]. Consequently, the straight-line drawing convention is o... |

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Citation Context ...y. In the example of Figure 1, the vertices are assigned to layers for semantic reasons. In other examples, the layer assignment is chosen to improve the readability of the drawing; see, for example, =-=[10]-=-. ffl Clustered graphs are graphs with recursive clustering structures which appear in many structured diagrams [24, 30, 31, 45]. For example, Figure 2 shows a relational diagram of some organizations... |

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Citation Context ...ye can follow a straight-line easily; Sugiyama [48] and Batini [51] list straightness of lines as an important aim for graph drawing algorithms. This intuition has been confirmed by human experiments =-=[41]-=-, [40]. Consequently, the straight-line drawing convention is one of many important modern aesthetic criteria [2] in graph drawing. More importantly, there are several general methods for drawing grap... |

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Citation Context ... hierarchical structures are involved e.g. PERT networks and organization charts [19, 10]. Clustered graphs are graphs with clustering structures (see Fig. 2) which appear in many structured diagrams =-=[18, 11, 12]-=-. A hierarchical graph is conventionally drawn with vertices of a layer on the same horizontal line, and arcs as curves monotonic in y direction (see Fig. 1). A hierarchical graph is hierarchical plan... |

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Citation Context ...e graph. Such algorithms aim to produce drawings which are easy to read and easy to remember. Many graph drawing algorithms have been designed, analyzed, tested and used in visualization systems [2], =-=[33]-=-. With increasing complexity of the information that we want to visualize, we need more structure on top of the classical graph model. Several extended graph models have been proposed [5], [26], [35],... |

67 | Drawing planar graphs using the canonical ordering
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Citation Context ... Feng, X. Lin, and H. Nagamochi et al. [8], [9]. More recently, efficient algorithms for planar straight-line drawings were developed by de Fraysseix et al. [23], Schnyder [46], Chrobak [10] and Kant =-=[32]-=-. These recent methods show that every planar graph admits a straight-line drawing in which each vertex is located at an integer grid point and the whole drawing uses O(n 2 ) area. In this paper we in... |

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Citation Context ...nt both to a lower-numbered vertex and a highernumbered vertex. Vertices s and t are called the source and the sink respectively. An st numbering of a biconnected graph can be computed in linear time =-=[16]-=-. An outline of our algorithm is described as follows. We need to generalize this notion to clustered graphs. Given a clustered graph C = (G; T ), an st numbering of the vertices of G such that the ve... |

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Citation Context ... 11, 23, 34, 39, 46], they all introduce bends to route the edges, and the basic problem of planar straight-line drawings has not been solved completely. It has been shown by di Battista and Tamassia =-=[2]-=- that every planar st-graph admits an upward drawing, that is, a drawing where all arcs are drawn as straight-line segments pointing upward. However, the problem for hierarchical graphs is different, ... |

56 |
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Citation Context ... or edge-region crossings. One of the basic graph drawing convention consists of representing edges as straight-line segments. For classical graphs, it has been shown independently by Fary [7], Stein =-=[17]-=-, and Wagner [21] that every planar graph admits a straight-line drawing without edge crossings. Tutte [20] proved that every 3-connected planar graph admits a planar straight-line drawing where all t... |

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Citation Context ...igure 1: An example of a hierarchical graph. ffl Hierarchical graphs are graphs with vertices assigned to layers. Hierarchical graphs appear in applications where hierarchical structures are involved =-=[23, 46]-=-. For example, Figure 1 shows a prerequisite diagram for subjects in a Bachelor's degree in Computer Science; an edge from a to b means that a is a prerequisite of b, and each layer represents a semes... |

47 |
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Citation Context ...ificance for drawing hierarchical graphs: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? While many algorithms have been developed to draw hierarchical graphs =-=[5, 10, 11, 23, 34, 39, 46]-=-, they all introduce bends to route the edges, and the basic problem of planar straight-line drawings has not been solved completely. It has been shown by di Battista and Tamassia [2] that every plana... |

47 |
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Citation Context ...ong to that cluster. A clustered graph is compound planar (c-planar) if it admits a drawing with no edge crossings or edge-region crossings. Algorithms for testing compound planarity are presented in =-=[21, 19]-=-. The hierarchical structure in hierarchical graphs imposes constraints on the y-coordinate, since all vertices of the same layer has to be drawn on exactly the same horizontal line. However, there ar... |

38 |
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Citation Context ...ical graphs: does every planar hierarchical graph admit a planar straight-line hierarchical drawing? While many algorithms have been developed to draw hierarchical graphs [7], [13], [15], [24], [38], =-=[43]-=-, [50], they all introduce bends to route the edges, and the basic problem of planar straight-line drawings has not been solved completely. It has been shown by Di Battista and Tamassia [4] that every... |

35 |
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Citation Context ...ry triconnected planar graph admits a planar straight-line drawing where all the face boundaries are drawn as convex polygons. Algorithms for such drawings have also been investigated by Chiba et al. =-=[6, 7]-=-. More recently, 3 efficient algorithms for planar straight-line drawings were developed by de Fraysseix et al. [22], Schnyder [41], Chrobak [8] and Kant [29]. These recent methods show that every pla... |

29 |
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Citation Context ...del relations in computing, and many systems for manipulating graphs have recently been developed. Examples include CASE tools [55] knowledge representation systems [31], software visualization tools =-=[54]-=- and VLSI design systems [26]. A graph drawing algorithm reads as input a combinatorial description of a graph, and produces as output a visual representation of the graph. Such algorithms aim to prod... |

25 | Algorithms for drawing clustered graphs
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Citation Context ...phs should address the following open problems: ffl We note that relaxing the straight-line constraints can give us polynomial area bounds both in hierarchical drawings and in convex cluster drawings =-=[18, 12, 13]-=-. In future work, we would like to investigate the trade-off between the number of bends and the area of the drawing. ffl The drawings of clustered graphs may lack vertical compaction because we use a... |

22 |
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Citation Context ...ry triconnected planar graph admits a planar straight-line drawing where all the face boundaries are drawn as convex polygons. Algorithms for such drawings have also been investigated by Chiba et al. =-=[6, 7]-=-. More recently, 3 efficient algorithms for planar straight-line drawings were developed by de Fraysseix et al. [22], Schnyder [41], Chrobak [8] and Kant [29]. These recent methods show that every pla... |

20 |
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Citation Context ...drawings have also been investigated by Chiba et al. [6, 7]. More recently, 3 efficient algorithms for planar straight-line drawings were developed by de Fraysseix et al. [22], Schnyder [41], Chrobak =-=[8]-=- and Kant [29]. These recent methods show that every planar graph admits a straight-line drawing in which each vertex is located at an integer grid point and the whole drawing uses O(n 2 ) area. In th... |

18 | An Algorithm for Drawing a Hierarchical Graph
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(Show Context)
Citation Context ...drawn on the same horizontal line and the layers should be an equal distance apart. A method to construct straight-line drawings of planar hierarchical graphs was presented by Eades, Lin and Tamassia =-=[14]-=-. They use a technique similar to that in [48] by Tutte, finding the position for every vertex in a global manner. In their algorithm, dummy vertices are added to transform an edge that spans more tha... |

17 |
Graphed: An interactive graph editor
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Citation Context ...ns consists of representing edges as straight-line segments. The straight-line drawing convention is widely used in visualization. Graph drawing systems such as the Graph Layout Toolkit [42], GraphEd =-=[26]-=- and Diagram Server [1] contain a module for creating straight-line drawings of classical graphs. Intuitively, the eye can follow a straight-line easily; Sugiyama [44] and Batini [47] list straightnes... |

17 |
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Citation Context ...is a sub-clustered graph of a connected and c-planar clustered graph. From Theorem 3, we can assume that we are given a connected clustered graph when drawing a c-planar clustered graph. According to =-=[13]-=-, a c-planar embedding of a connected clustered graph can be found in linear time. In the rest of the paper, we assume there are no degenerated clusters, that is, every nonleaf node of T has at least ... |

17 |
Analysis of Algorithms for Drawing Graphs
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Citation Context ...og n) is optimal for this type of problems. The drawings produced by our algorithms may require exponential area. This is justified by the area lower bounds for these drawing conventions presented in =-=[14, 8]-=-. Relaxing the straight-line constraints can give us polynomial area bounds [4]. For future work on clustered graphs, other drawing conventions such as polyline rectangular cluster drawings, and also ... |

15 | Drawing clustered graphs on an orthogonal grid
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(Show Context)
Citation Context ...phs should address the following open problems: ffl We note that relaxing the straight-line constraints can give us polynomial area bounds both in hierarchical drawings and in convex cluster drawings =-=[18, 12, 13]-=-. In future work, we would like to investigate the trade-off between the number of bends and the area of the drawing. ffl The drawings of clustered graphs may lack vertical compaction because we use a... |

14 |
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Citation Context ...ing edges as straight-line segments. The straight-line drawing convention is widely used in visualization. Graph drawing systems such as the Graph Layout Toolkit [42], GraphEd [26] and Diagram Server =-=[1]-=- contain a module for creating straight-line drawings of classical graphs. Intuitively, the eye can follow a straight-line easily; Sugiyama [44] and Batini [47] list straightness of lines as an import... |

13 |
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Citation Context ...ong to that cluster. A clustered graph is compound planar (c-planar) if it admits a drawing with no edge crossings or edge-region crossings. Algorithms for testing compound planarity are presented in =-=[25, 22, 20]-=-. The hierarchical structure in hierarchical graphs imposes constraints on the y-coordinate, since all vertices of the same layer have to be drawn on exactly the same horizontal line. However, there a... |

12 |
Planarization of clustered graphs
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(Show Context)
Citation Context ...raphs should address the following open problems: • We note that relaxing the straight-line constraints can give us polynomial area bounds both in hierarchical drawings and in convex cluster drawings =-=[1, 19, 13, 14]-=-. In future work, we would like to investigate the trade-off between the number of bends and the area of the drawing. • The drawings of clustered graphs may lack vertical compaction because we use an ... |

10 | Pitfalls of using PQ-trees in automatic graph drawing
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Citation Context ...rves monotonic in the y direction. A hierarchical graph is hierarchical planar (h-planar) if it admits a drawing without edge crossings. Algorithms for testing hierarchical planarity are presented in =-=[29]-=-. For a clustered graph, the clustering structure is represented by a closed curve that defines a region. The region contains the drawing of all the vertices which belong to that cluster. A clustered ... |

9 |
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Citation Context ...aph Layout Toolkit [44], GraphEd [28] and Diagram Server [3] contain a module for creating straight-line drawings of classical graphs. Intuitively, the eye can follow a straight-line easily; Sugiyama =-=[48]-=- and Batini [51] list straightness of lines as an important aim for graph drawing algorithms. This intuition has been confirmed by human experiments [41], [40]. Consequently, the straight-line drawing... |

5 | Clustered graphs and C-planarity
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Citation Context ...ong to that cluster. A clustered graph is compound planar (c-planar) if it admits a drawing with no edge crossings or edge-region crossings. Algorithms for testing compound planarity are presented in =-=[21, 19]-=-. The hierarchical structure in hierarchical graphs imposes constraints on the y-coordinate, since all vertices of the same layer has to be drawn on exactly the same horizontal line. However, there ar... |

5 |
Building Interactive Digram Applications
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Citation Context ... hierarchical structures are involved e.g. PERT networks and organization charts [19, 10]. Clustered graphs are graphs with clustering structures (see Fig. 2) which appear in many structured diagrams =-=[18, 11, 12]-=-. A hierarchical graph is conventionally drawn with vertices of a layer on the same horizontal line, and arcs as curves monotonic in y direction (see Fig. 1). A hierarchical graph is hierarchical plan... |

5 |
Visualizing Abstract Objects and Relations. World Scientific
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Citation Context ...ces. Graphs are commonly used to model relations in computing, and many systems for manipulating graphs have recently been developed. Examples include CASE tools [55] knowledge representation systems =-=[31]-=-, software visualization tools [54] and VLSI design systems [26]. A graph drawing algorithm reads as input a combinatorial description of a graph, and produces as output a visual representation of the... |

4 | Orthogonal grid drawing of clustered graphs
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(Show Context)
Citation Context ... may require exponential area. This is justified by the area lower bounds for these drawing conventions presented in [14, 8]. Relaxing the straight-line constraints can give us polynomial area bounds =-=[4]-=-. For future work on clustered graphs, other drawing conventions such as polyline rectangular cluster drawings, and also nonplanar drawings will be studied. For clustered graphs, we only ensure that c... |

4 |
Methods for computer display and manipulation of graphs and the corresponding algorithms
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(Show Context)
Citation Context ...given a c-planar connected clustered graph C = (G; T ) with a c-planar embedding. Roughly speaking, our algorithm works as follows. First, we triangulate G (including triangulating the external face) =-=[16]-=-; then compute an st numbering 1 of the vertices of G such that the vertices that belong to the same cluster are numbered consecutively. We call this numbering c-st numbering. This numbering gives us ... |

3 |
Hiroshi Nagamochi. Drawing clustered graphs on an orthogonal grid
- Eades, Feng, et al.
- 1999
(Show Context)
Citation Context ...ch cluster is a convex hull of points in the cluster can be constructed in O(n + D) time, where D = O(n 2 ) is the total size of convex polygons for clusters in the drawing. Note that as we showed in =-=[14]-=-, the results in this section, combining with the techniques in [14], yield an O(n) time drawing algorithm. The algorithm can always provide a planar straight-line drawing for a c-planar clustered gra... |