## ON-LINE PLANARITY TESTING (1996)

Citations: | 24 - 2 self |

### BibTeX

@MISC{Battista96on-lineplanarity,

author = {Giuseppe Di Battista and Roberto Tamassia},

title = {ON-LINE PLANARITY TESTING},

year = {1996}

}

### OpenURL

### Abstract

The on-line planarity-testing problem consists of performing the following operations on a planar graph G: (i) testing if a new edge can be added to G so that the resulting graph is itself planar; (ii) adding vertices and edges such that planarity is preserved. An efficient technique for online planarity testing of a graph is presented that uses O(n) space and supports tests and insertions of vertices and edges in O(log n) time, where n is the current number of vertices of G. The bounds for tests and vertex insertions are worst-case and the bound for edge insertions is amortized. We also present other applications of this technique to dynamic algorithms for planar graphs.

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Citation Context ...O(n) space and O(log n) query and update time exist for some classes of planar digraphs [11, 34, 54, 56]. The related problem of maintaining a topological ordering of an acyclic digraph is studied in =-=[1]-=-.958 GIUSEPPE DI BATTISTA AND ROBERTO TAMASSIA A shortest-path query in a digraph asks for the length of a shortest path between two vertices. Fully dynamic data structures for shortest-path queries ... |

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Citation Context ...s on planar graphs include algorithms for parallel planarity testing [37, 46], embedding [4, 62], drawing [10, 13, 19, 51], reachability [36, ,54, 57], shortest paths [22], and minimum spanning trees =-=[15, 21]-=-. Previous work on dynamic graph algorithms is surveyed in 2. The technique of [53] is a first step toward on-line planarity testing. Namely, it solves the restricted problem of maintaining a planar e... |

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Citation Context ...f an efficient algorithm for on-line planarity testing has been an elusive goal. Recent results on planar graphs include algorithms for parallel planarity testing [37, 46], embedding [4, 62], drawing =-=[10, 13, 19, 51]-=-, reachability [36, ,54, 57], shortest paths [22], and minimum spanning trees [15, 21]. Previous work on dynamic graph algorithms is surveyed in 2. The technique of [53] is a first step toward on-line... |

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Citation Context ...ime per insertion for dense graphs [2, 39]. For series-parallel digraphs with weighted edges, there exists a fully dynaxnic O(n)-space data structure that supports queries and updates in O(logn) time =-=[9]-=-. This data structure also maintains a maximum flow within the same time bounds. The dynamic maintenance of minimum spanning trees has the interesting property that, after an update operation consisti... |

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Citation Context ...the development of an efficient algorithm for on-line planarity testing has been an elusive goal. Recent results on planar graphs include algorithms for parallel planarity testing [37, 46], embedding =-=[4, 62]-=-, drawing [10, 13, 19, 51], reachability [36, ,54, 57], shortest paths [22], and minimum spanning trees [15, 21]. Previous work on dynamic graph algorithms is surveyed in 2. The technique of [53] is a... |

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Citation Context ...same performance is obtained for biconnected components [42, 60], triconnected components [11, 42], and tbur-connected components [35]. Semidynamic techniques supporting deletions only are studied in =-=[17, 47]-=-. In a fully dynamic environment, the connected components of a general graph can be maintained in time O(v/) per update operation [21], while the biconnected components of a planar graph can be maint... |

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Citation Context ...s whether there is a directed path between two vertices. For general digraphs, there exist insertions-only semidynamic data structures with O(n) space, O(1) query time, and O(n) amortized update time =-=[6, 32, 43]-=-. The same performance is achieved for deletions only in acyclic digraphs [6, 33]. Fully dynamic data structures with O(n) space and O(log n) query and update time exist for some classes of planar dig... |

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Citation Context ...vely, denote the number of vertices and edges of the graph being considered. A general lower-bound technique for incremental algorithms, with applications to dynamic graph algorithms, is discussed in =-=[3]-=-. A teachability query in a digraph asks whether there is a directed path between two vertices. For general digraphs, there exist insertions-only semidynamic data structures with O(n) space, O(1) quer... |

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Citation Context ...f an efficient algorithm for on-line planarity testing has been an elusive goal. Recent results on planar graphs include algorithms for parallel planarity testing [37, 46], embedding [4, 62], drawing =-=[10, 13, 19, 51]-=-, reachability [36, ,54, 57], shortest paths [22], and minimum spanning trees [15, 21]. Previous work on dynamic graph algorithms is surveyed in 2. The technique of [53] is a first step toward on-line... |

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Citation Context ...ientation if and only if it is planarly st-biconnectible [38], i.e., the graph obtained from G by adding the edge (s, t) is planar and biconnected. A planar storientation can be computed in O(n) time =-=[18]-=-. 4. Tests. In this section, we consider the problem of performing operation Test on a biconnected planar graph with n vertices. We assume that the graph has been oriented into a planar st-graph such ... |

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Citation Context ...same performance is achieved for deletions only in acyclic digraphs [6, 33]. Fully dynamic data structures with O(n) space and O(log n) query and update time exist for some classes of planar digraphs =-=[11, 34, 54, 56]-=-. The related problem of maintaining a topological ordering of an acyclic digraph is studied in [1].958 GIUSEPPE DI BATTISTA AND ROBERTO TAMASSIA A shortest-path query in a digraph asks for the lengt... |

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Citation Context ...USEPPE DI BATTISTA AND ROBERTO TAMASSIA edge ordering is the planar subgraph S of G with maximum k(S). Several heuristics have been developed for computing maximum-weight planar subgraphs; see, e.g., =-=[14, 41]-=-. There is experimental evidence that if the edges are sorted by decreasing weight, a lexicographically maximum planar subgraph provides a good approximation of a maximum-weight planar subgraph [14]. ... |

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Citation Context ...f an efficient algorithm for on-line planarity testing has been an elusive goal. Recent results on planar graphs include algorithms for parallel planarity testing [37, 46], embedding [4, 62], drawing =-=[10, 13, 19, 51]-=-, reachability [36, ,54, 57], shortest paths [22], and minimum spanning trees [15, 21]. Previous work on dynamic graph algorithms is surveyed in 2. The technique of [53] is a first step toward on-line... |

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Citation Context ...same performance is achieved for deletions only in acyclic digraphs [6, 33]. Fully dynamic data structures with O(n) space and O(log n) query and update time exist for some classes of planar digraphs =-=[11, 34, 54, 56]-=-. The related problem of maintaining a topological ordering of an acyclic digraph is studied in [1].958 GIUSEPPE DI BATTISTA AND ROBERTO TAMASSIA A shortest-path query in a digraph asks for the lengt... |

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Citation Context ...tion of two split components of the same split pair. For example, Fig. 5 shows the planar embedding obtained from that of Fig. 1 by means of two swap operations and one flip operation. LEMMA 4.2 (see =-=[12]-=-). Given a pair of embeddings F’ and F" of a planar st-graph G with the edge (s, t), F" can be obtained from F’ by means of a sequence of O(n) reverse and swap operations. By Lemma 4.2, the decomposit... |

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Citation Context ...s on planar graphs include algorithms for parallel planarity testing [37, 46], embedding [4, 62], drawing [10, 13, 19, 51], reachability [36, ,54, 57], shortest paths [22], and minimum spanning trees =-=[15, 21]-=-. Previous work on dynamic graph algorithms is surveyed in 2. The technique of [53] is a first step toward on-line planarity testing. Namely, it solves the restricted problem of maintaining a planar e... |

1 |
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Citation Context ...en an elusive goal. Recent results on planar graphs include algorithms for parallel planarity testing [37, 46], embedding [4, 62], drawing [10, 13, 19, 51], reachability [36, ,54, 57], shortest paths =-=[22]-=-, and minimum spanning trees [15, 21]. Previous work on dynamic graph algorithms is surveyed in 2. The technique of [53] is a first step toward on-line planarity testing. Namely, it solves the restric... |

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Citation Context ...le the biconnected components of a planar graph can be maintained in O(n2/a) time using O(n) space [25]. The related problems of maintaining the two- and three-edgeconnected components are studied in =-=[23, 24, 60]-=-. 3. Preliminaries. We assume that the reader is familiar with graph terminology and basic properties of planar graphs (see, e.g., [40]). Throughout this paper, denotes the number of vertices of the p... |

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Citation Context ...le the biconnected components of a planar graph can be maintained in O(n2/a) time using O(n) space [25]. The related problems of maintaining the two- and three-edgeconnected components are studied in =-=[23, 24, 60]-=-. 3. Preliminaries. We assume that the reader is familiar with graph terminology and basic properties of planar graphs (see, e.g., [40]). Throughout this paper, denotes the number of vertices of the p... |