## Clustered Graphs and C-planarity (1995)

Venue: | In 3rd Annual European Symposium on Algorithms (ESA’95), LNCS 979 |

Citations: | 5 - 2 self |

### BibTeX

@TECHREPORT{Feng95clusteredgraphs,

author = {Qing-wen Feng and Peter Eades and Robert F. Cohen},

title = {Clustered Graphs and C-planarity},

institution = {In 3rd Annual European Symposium on Algorithms (ESA’95), LNCS 979},

year = {1995}

}

### OpenURL

### Abstract

In this paper, we introduce a new graph model known as clustered graphs, i.e. graphs with recursive clustering structures. This graph model has many applications in informational and mathematical sciences. In particular, we study C-planarity of clustered graphs. Given a clustered graph, the C-planarity testing problem is to determine whether the clustered graph can be drawn without edge crossings, or edge-region crossings. In this paper, we present efficient algorithms for testing C-planarity and finding C-planar embeddings of clustered graphs. 1 Introduction Representing information visually, or by drawing graphs can greatly improve the effectiveness of user interfaces in many relational information systems [12, 17, 18, 5]. Developing algorithms for drawing graphs automatically and efficiently has become the interest of research for many computer scientists. Research in this area has been very active for the last decade. A recent survey citelabel13new of literature in this area inclu...

### Citations

2438 |
The Design and Analysis of Computer Algorithms
- Aho, Hopcroft, et al.
- 1974
(Show Context)
Citation Context ...given graph G with n vertices and m edges, the PQ-tree planarity testing algorithm requires at most O(n) steps. There are algorithms requiring O(n + m) steps to find biconnected components of a graph =-=[1]-=-, and to generate st-numbering for each biconnected component [8]. Note that graph G is updated throughout our algorithm. At nodesof T , the vertices in G that serve as hubs of wheels have no connecti... |

544 | On Visual Formalisms
- Harel
- 1988
(Show Context)
Citation Context ...graphs tend to be insufficient for modeling the information. This has motivated the development of more powerful graph models, e.g. hypergraphs [2], compound digraphs [21], cigraphs [15] and higraphs =-=[10]-=-. Although these graph models has provided us a high level of abstraction and can be applied to a wide range of applications, automatic layout facilities for these graphs seem hard to develop. Only he... |

468 |
Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms
- Booth, Lueker
- 1976
(Show Context)
Citation Context ... example, the problem of minimizing edge crossings is proved to be NP-hard [9, 7]. However, efficient algorithms for testing whether a graph is planar (i.e. can be drawn without edge crossings) exist =-=[11, 16, 3, 6]-=-. Planarity issues relating to the more powerful graph models mentioned above have not been studied. In this paper, we introduce C-planarity, the planarity of clustered graphs. In a drawing of a clust... |

226 | Tarjan, Efficient planarity testing
- Hopcroft, E
- 1974
(Show Context)
Citation Context ... example, the problem of minimizing edge crossings is proved to be NP-hard [9, 7]. However, efficient algorithms for testing whether a graph is planar (i.e. can be drawn without edge crossings) exist =-=[11, 16, 3, 6]-=-. Planarity issues relating to the more powerful graph models mentioned above have not been studied. In this paper, we introduce C-planarity, the planarity of clustered graphs. In a drawing of a clust... |

208 |
Crossing number is NP-complete
- MR, DS
- 1983
(Show Context)
Citation Context ...ganization [14], software visualization [22] and VLSI design [10]. Planarity is a much studied area for classical graphs. For example, the problem of minimizing edge crossings is proved to be NP-hard =-=[9, 7]-=-. However, efficient algorithms for testing whether a graph is planar (i.e. can be drawn without edge crossings) exist [11, 16, 3, 6]. Planarity issues relating to the more powerful graph models menti... |

112 |
A linear algorithm for embedding planar graphs using PQ-trees
- Chiba, Nishizeki, et al.
- 1985
(Show Context)
Citation Context ...t to algorithm CPEmbed is a connected clustered graph. The algorithm returns a C-planar embedding if the input clustered graph is C-planar; otherwise, returns an empty embedding. The algorithm PEmbed =-=[4]-=- replaces algorithm PT in our algorithm CPEmbed. The algorithm PEmbed tests the planarity of a graph and finds a planar embedding if the graph is planar. It uses the same PQ-tree technique as algorith... |

104 |
Cederbaum I. An algorithm for planarity testing of graphs
- Lempel, Even
- 1967
(Show Context)
Citation Context ... example, the problem of minimizing edge crossings is proved to be NP-hard [9, 7]. However, efficient algorithms for testing whether a graph is planar (i.e. can be drawn without edge crossings) exist =-=[11, 16, 3, 6]-=-. Planarity issues relating to the more powerful graph models mentioned above have not been studied. In this paper, we introduce C-planarity, the planarity of clustered graphs. In a drawing of a clust... |

69 |
Visualizing and querying software structures
- Consens, Mendelzon, et al.
- 1992
(Show Context)
Citation Context ... embeddings of clustered graphs. 1 Introduction Representing information visually, or by drawing graphs can greatly improve the effectiveness of user interfaces in many relational information systems =-=[12, 17, 18, 5]-=-. Developing algorithms for drawing graphs automatically and efficiently has become the interest of research for many computer scientists. Research in this area has been very active for the last decad... |

69 |
Visualization of structural information: Automatic drawing of compound digraphs
- Sugiyama, Misue
- 1991
(Show Context)
Citation Context ... and more complicated, classical graphs tend to be insufficient for modeling the information. This has motivated the development of more powerful graph models, e.g. hypergraphs [2], compound digraphs =-=[21]-=-, cigraphs [15] and higraphs [10]. Although these graph models has provided us a high level of abstraction and can be applied to a wide range of applications, automatic layout facilities for these gra... |

62 |
Computing an st-numbering
- Even, Tarjan
- 1976
(Show Context)
Citation Context ...testing algorithm requires at most O(n) steps. There are algorithms requiring O(n + m) steps to find biconnected components of a graph [1], and to generate st-numbering for each biconnected component =-=[8]-=-. Note that graph G is updated throughout our algorithm. At nodesof T , the vertices in G that serve as hubs of wheels have no connection with the rest of the graph except the vertices on the correspo... |

34 |
On an edge crossing problem
- Eades, McKay, et al.
- 1986
(Show Context)
Citation Context ...ganization [14], software visualization [22] and VLSI design [10]. Planarity is a much studied area for classical graphs. For example, the problem of minimizing edge crossings is proved to be NP-hard =-=[9, 7]-=-. However, efficient algorithms for testing whether a graph is planar (i.e. can be drawn without edge crossings) exist [11, 16, 3, 6]. Planarity issues relating to the more powerful graph models menti... |

31 |
Rigi - A Model for Software System Construction, Integration, and Evolution based on Module Interfaces Specifications
- Müller
- 1986
(Show Context)
Citation Context ... embeddings of clustered graphs. 1 Introduction Representing information visually, or by drawing graphs can greatly improve the effectiveness of user interfaces in many relational information systems =-=[12, 17, 18, 5]-=-. Developing algorithms for drawing graphs automatically and efficiently has become the interest of research for many computer scientists. Research in this area has been very active for the last decad... |

29 |
The state of the art of visual languages for visualization
- Williams, Rasure, et al.
- 1992
(Show Context)
Citation Context ...see Fig. 1). This clustering structure can be used to model information in many areas, such as software engineering [23], knowledge representation [13], idea organization [14], software visualization =-=[22]-=- and VLSI design [10]. Planarity is a much studied area for classical graphs. For example, the problem of minimizing edge crossings is proved to be NP-hard [9, 7]. However, efficient algorithms for te... |

22 |
Visualizing Abstract Objects and Relations
- Kamada
- 1989
(Show Context)
Citation Context ...s, i.e. graphs with recursive clustering structures (see Fig. 1). This clustering structure can be used to model information in many areas, such as software engineering [23], knowledge representation =-=[13]-=-, idea organization [14], software visualization [22] and VLSI design [10]. Planarity is a much studied area for classical graphs. For example, the problem of minimizing edge crossings is proved to be... |

20 |
A depth-first search characterization of planarity
- Fraysseix, Rosenstiehl
- 1982
(Show Context)
Citation Context |

13 |
Drawing ranked digraphs with recursive clusters, in
- North
- 1993
(Show Context)
Citation Context ... applied to a wide range of applications, automatic layout facilities for these graphs seem hard to develop. Only heuristic algorithms for hierarchical layout of compound digraphs have been developed =-=[21, 20]-=-. In this paper, we introduce a practical and simple model called clustered graphs, i.e. graphs with recursive clustering structures (see Fig. 1). This clustering structure can be used to model inform... |

10 |
KJ method: A scientific approach to problem solving
- Kawakita
- 1975
(Show Context)
Citation Context ...rsive clustering structures (see Fig. 1). This clustering structure can be used to model information in many areas, such as software engineering [23], knowledge representation [13], idea organization =-=[14]-=-, software visualization [22] and VLSI design [10]. Planarity is a much studied area for classical graphs. For example, the problem of minimizing edge crossings is proved to be NP-hard [9, 7]. However... |

9 |
An overview of diagram based idea organizer: D-ABDUCTOR
- Misue, Sugiyama
- 1993
(Show Context)
Citation Context ... embeddings of clustered graphs. 1 Introduction Representing information visually, or by drawing graphs can greatly improve the effectiveness of user interfaces in many relational information systems =-=[12, 17, 18, 5]-=-. Developing algorithms for drawing graphs automatically and efficiently has become the interest of research for many computer scientists. Research in this area has been very active for the last decad... |

5 |
Building Interactive Digram Applications
- Lai
- 1993
(Show Context)
Citation Context ...icated, classical graphs tend to be insufficient for modeling the information. This has motivated the development of more powerful graph models, e.g. hypergraphs [2], compound digraphs [21], cigraphs =-=[15]-=- and higraphs [10]. Although these graph models has provided us a high level of abstraction and can be applied to a wide range of applications, automatic layout facilities for these graphs seem hard t... |

3 |
and N.Chiba. Planar Graphs : Theory and algorithms. Annals of discrete mathematics 32
- Nishizeki
- 1988
(Show Context)
Citation Context ...s s and t are called the source and the sink respectively. We need the following lemma to understand the PQ-tree planarity testing algorithm and also to show the correctness of our algorithm. Lemma 1 =-=[19]-=- Suppose that a graph G is a biconnected and st-numbered planar graph. Let G k = (V k ; E k ) be the subgraph of G induced by vertices V k = f1; 2; : : : ; kg, 1sksn. If edge (s; t) is drawn on the bo... |