## C-planarity of extrovert clustered graphs (2005)

Venue: | In Graph Drawing |

Citations: | 6 - 1 self |

### BibTeX

@INPROCEEDINGS{Goodrich05c-planarityof,

author = {Michael T. Goodrich and George S. Lueker and Jonathan Z. Sun},

title = {C-planarity of extrovert clustered graphs},

booktitle = {In Graph Drawing},

year = {2005},

pages = {211--222}

}

### OpenURL

### Abstract

Abstract. A clustered graph has its vertices grouped into clusters in a hierarchical way via subset inclusion, thereby imposing a tree structure on the clustering relationship. The c-planarity problem is to determine if such a graph can be drawn in a planar way, with clusters drawn as nested regions and with each edge (drawn as a curve between vertex points) crossing the boundary of each region at most once. Unfortunately, as with the graph isomorphism problem, it is open as to whether the cplanarity problem is NP-complete or in P. In this paper, we show how to solve the c-planarity problem in polynomial time for a new class of clustered graphs, which we call extrovert clustered graphs. This class is quite natural (we argue that it captures many clustering relationships that are likely to arise in practice) and includes the clustered graphs tested in previous work by Dahlhaus, as well as Feng, Eades, and Cohen. Interestingly, this class of graphs does not include, nor is it included by, a class studied recently by Gutwenger et al.; therefore, this paper offers an alternative advancement in our understanding of the efficient drawability of clustered graphs in a planar way. Our testing algorithm runs in O(n 3) time and implies an embedding algorithm with the same time complexity. 1

### Citations

477 |
Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J o u d of Chtnputet and Syskms Science 13
- Booth, Lueker
- 1976
(Show Context)
Citation Context ...rcuit, we might want to piece each functional module together in a hierarchical way. While the problem of determining if a given graph is planar is well-known to be solvable in linear time (e.g., see =-=[3, 14, 17]-=-), the general c-planarity problem x1 x3 x6 �1 �2 �1 x4 x5 �3 x2 �2 x1 x1 x6 x3 x3 x6 (a) (b) (c) Fig.1. (a) A c-graph C with 5 clusters µ1 = {x1}, µ2 = {x2}, µ3 = {x3,x4,x5,x6}, ν1 = {x3,x5} and ν2 =... |

232 | Efficient Planarity Testing
- Hopcroft, Tarjan
- 1974
(Show Context)
Citation Context ...rcuit, we might want to piece each functional module together in a hierarchical way. While the problem of determining if a given graph is planar is well-known to be solvable in linear time (e.g., see =-=[3, 14, 17]-=-), the general c-planarity problem x1 x3 x6 �1 �2 �1 x4 x5 �3 x2 �2 x1 x1 x6 x3 x3 x6 (a) (b) (c) Fig.1. (a) A c-graph C with 5 clusters µ1 = {x1}, µ2 = {x2}, µ3 = {x3,x4,x5,x6}, ν1 = {x3,x5} and ν2 =... |

105 | An algorithm for planarity testing of graphs - LEMPEL, EVEN, et al. - 1967 |

83 | Multilevel visualization of clustered graphs
- Eades, Feng
- 1996
(Show Context)
Citation Context ...ge or splitting one cluster in this c-graph will make it c-planar, as illustrated in (b) and (c). Although a number of papers have addressed the problem of how to draw a c-planar c-graph in the plane =-=[1,6,7,8,9,10]-=-, very little progress has been made in testing the c-planarity of a given c-graph. Previous work provides effective tests only for a few special classes of c-graphs [4,5,12,13]. In this paper we defi... |

79 | On-line planarity testing
- Battista, Tamassia
- 1989
(Show Context)
Citation Context ...nding on where we start, and whether we read clockwise or counterclockwise, we can obtain various permutations; we will say these permutations are circularly equivalent. For example, the permutations =-=(3, 5, 2, 4, 1, 6)-=-, (2, 4, 1, 6, 3, 5), and (5, 3, 6, 1, 4, 2) are circularly equivalent. We call an equivalence class of this relation a circular permutation. Wesayany element of the equivalence class is a representat... |

63 | Tarjan, “Computing an st-numbering - Even, E - 1976 |

59 | Straight-line drawing algorithms for hierarchical graphs and clustered graphs
- Eades, Feng, et al.
- 1996
(Show Context)
Citation Context ...ge or splitting one cluster in this c-graph will make it c-planar, as illustrated in (b) and (c). Although a number of papers have addressed the problem of how to draw a c-planar c-graph in the plane =-=[1,6,7,8,9,10]-=-, very little progress has been made in testing the c-planarity of a given c-graph. Previous work provides effective tests only for a few special classes of c-graphs [4,5,12,13]. In this paper we defi... |

49 | Navigating Clustered Graph using ForceDirected Methods
- Eades
(Show Context)
Citation Context ...ge or splitting one cluster in this c-graph will make it c-planar, as illustrated in (b) and (c). Although a number of papers have addressed the problem of how to draw a c-planar c-graph in the plane =-=[1,6,7,8,9,10]-=-, very little progress has been made in testing the c-planarity of a given c-graph. Previous work provides effective tests only for a few special classes of c-graphs [4,5,12,13]. In this paper we defi... |

35 | PC-trees and circular-ones arrangements
- Hsu, McConnell
(Show Context)
Citation Context ... a circular permutation π if it is consecutive in any representative of π. Informally, this means that the elements of S appear consecutively around the circle. 2.3 PC-Trees and PC-Reduction PC-trees =-=[15]-=- provide an elegant structure that both simplifies PQ-trees and allows convenient operations on circular permutations. A PC-tree is an unrooted tree with two types of internal nodes, P-nodes and C-nod... |

15 | Drawing clustered graphs on an orthogonal grid
- Eades, Feng, et al.
- 1999
(Show Context)
Citation Context |

14 | Linear time algorithm to recognize clustered planar graphs and its parallelization
- Dahlhaus
- 1998
(Show Context)
Citation Context ...-graph in the plane [1,6,7,8,9,10], very little progress has been made in testing the c-planarity of a given c-graph. Previous work provides effective tests only for a few special classes of c-graphs =-=[4,5,12,13]-=-. In this paper we define and test a new class of c-graphs, which generalizes the result in [5, 12] but is not comparable with [4,13]. The general problem is still open. So far the testing problem and... |

13 | Advances in cplanarity testing of clustered graphs - GUTWENGER, JÜNGER, et al. - 2002 |

12 |
Planarization of clustered graphs
- Battista, Didimo, et al.
- 2001
(Show Context)
Citation Context |

10 | Clustering cycles into cycles of clusters
- Cortese, Battista, et al.
(Show Context)
Citation Context ...-graph in the plane [1,6,7,8,9,10], very little progress has been made in testing the c-planarity of a given c-graph. Previous work provides effective tests only for a few special classes of c-graphs =-=[4,5,12,13]-=-. In this paper we define and test a new class of c-graphs, which generalizes the result in [5, 12] but is not comparable with [4,13]. The general problem is still open. So far the testing problem and... |

10 | Planarity-preserving clustering and embedding for large planar graphs
- Duncan, Goodrich, et al.
(Show Context)
Citation Context |

6 |
Planar Graphs: Theory and Algorithms, volume 32 of Annals of Discrete Mathematics
- Nishizeki, Chiba
- 1988
(Show Context)
Citation Context ...rcuit, we might want to piece each functional module together in a hierarchical way. While the problem of determining if a given graph is planar is well-known to be solvable in linear time (e.g., see =-=[3, 14, 17]-=-), the general c-planarity problem x1 x3 x6 �1 �2 �1 x4 x5 �3 x2 �2 x1 x1 x6 x3 x3 x6 (a) (b) (c) Fig.1. (a) A c-graph C with 5 clusters µ1 = {x1}, µ2 = {x2}, µ3 = {x3,x4,x5,x6}, ν1 = {x3,x5} and ν2 =... |

5 | Clustered graphs and C-planarity
- Feng, Eades, et al.
- 1995
(Show Context)
Citation Context ...usion is to draw the clustered graph in a planar way. Deciding if such a drawing is possible is one of the most interesting problems involving clustered graphs, and was posed by Feng, Eades and Cohen =-=[12]-=-. Formally, this problem, which is called the c-planarity problem,asksifCcan be drawn (or embedded) in the plane satisfying the following criteria: 1. There is no crossing between any edges of the und... |