## 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...

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |

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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... |