## ON-LINE PLANARITY TESTING (1996)

Citations: | 25 - 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.

### Citations

11412 |
Computers and Intractability. A Guide to the Theory of NP-Completeness. edited by W.H
- Garey, Johnson
- 1997
(Show Context)
Citation Context ...es of G, a maximum-weight planar subgraph of G is a planar subgraph of G with the maximum total edge weight. Finding a maximumweight planar subgraph is NP-hard even if all the edges have unit weights =-=[26]-=-. Given an ordering e0,..., e,-i of the edges of G, each subgraph S of G is identified by an integer k(S) (brn-l"’" b0)2 such that bit b 1 if and only if edge e is in S. The lexicographically maximum ... |

1244 |
Graph Theory
- Harary
- 1965
(Show Context)
Citation Context ...) of G, a C-node for each cutvertex of (, and edges connecting984 GIUSEPPE DI BATTISTA AND ROBERTO TAMASSIA each B-node # to the C-nodes associated with the cutvertices in the block of # (see, e.g., =-=[28]-=-). The block-cutvertex tree was previously used in [53, 60] for maintaining biconnected components. We construct an augmented block-cutvertex tree B for G as follows (see Fig. 19). We root B at an arb... |

341 |
Fast algorithms for finding nearest common ancestors
- Harel, Tarjan
- 1984
(Show Context)
Citation Context ...x a pointer to its proper allocation node in T. Hence step 1 takes O(1) time. We equip tree T with a data structure which uses linear space and supports least common ancestor queries in constant time =-=[29, 50]-=-. Hence step 2 takes O(1) time. Concerning step 3, we set up the following data structures. Each node of T has a pointer to the corresponding virtual edge in the skeleton of its parent. We mark all th... |

185 |
Dividing a graph into triconnected components
- Hopcroft, Tarjan
- 1973
(Show Context)
Citation Context ...e planar st-graph of Fig. 1. Our definition of decomposition tree is a variation of the one given in [4] and is closely related to the decomposition of biconnected graphs into triconnected components =-=[30]-=-. (0,17) (1,2) (2,3) (7,8) (8,17) (1,9) (9,17) (1,11) (11,12) (12,15) a8 (15,16) (: (12,17) (16,17) (3,4) (3,6) (4,7) (6,7) (1,10) (10,12) (4,5) (5,6) (11,18)(18,13)(13,19) (19,15) (11,14) (14,15) FIG... |

112 | A linear algorithm for embedding planar graphs using PQ-trees - Chiba, Nishizeki, et al. - 1985 |

71 |
Incremental evaluation of computational circuits
- Alpern, Hoover, et al.
- 1990
(Show Context)
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 ... |

67 |
Incremental algorithms for minimal length paths
- Ausiello, Italiano, et al.
- 1991
(Show Context)
Citation Context ...phs with unit edge lengths use O(n) space and have constant query time; the total time to process all edge insertions is O(na log n), which amortizes to O(n log n) time 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 ... |

66 | Corrigendum (Maintenance of a minimum spanning forest in a dynamic plane graph
- Eppstein, Italiano, et al.
- 1993
(Show Context)
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... |

46 |
Small sets supporting Fáry embeddings of planar graphs
- Fraysseix, Pach, et al.
- 1988
(Show Context)
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... |

45 | Updating Distances in Dynamic Graphs - Even, Gazit - 1985 |

44 | Dynamic trees and dynamic point location
- Goodrich, Tamassia
- 1991
(Show Context)
Citation Context ...tive closure. In this section, we study the maintenance of reachability information in planar st-graphs, an important class of planar digraphs that find several applications in computational geometry =-=[27, 44, 45]-=- and graph layout [10, 13]. Queries are of the following type: "Is there a directed path from vl to v27" This problem was previously best solved using a data structure for general digraphs with O(n2) ... |

35 |
Amortized efficiency of a path retrieval data structure
- Italiano
- 1986
(Show Context)
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... |

27 |
Dynamic expression trees and their applications
- Cohen, Tamassia
- 1991
(Show Context)
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... |

26 | On the complexity of covering vertices by faces in a planar graph
- Bienstock, Monma
- 1988
(Show Context)
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... |

26 |
An on-line edge deletion problem
- Even, Shiloach
- 1981
(Show Context)
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... |

19 |
A Data Structure for Arc Insertion and Regular Path Finding
- Buchsbaum, Kanellakis, et al.
- 1991
(Show Context)
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... |

12 |
Proving relative lower bounds for incremental algorithms
- Berman, Paull, et al.
- 1990
(Show Context)
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... |

9 |
Area requirement and symmetry display in drawing graphs
- DiBattista, Tamassia, et al.
- 1989
(Show Context)
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... |

9 |
Corrigendum: “Computing an st-numbering” (Theoret
- Even, Tarjan
- 1976
(Show Context)
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 ... |

3 |
Dynamic Data Structures for Series-Parallel Graphs
- Italiano, Marchetti-Spaccamela, et al.
- 1989
(Show Context)
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... |

2 |
An efficient heuristic for identifying a maximal weight planar subgraph
- EADES, FOULDS, et al.
- 1990
(Show Context)
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]. ... |

1 | Testing for the consecutive ones property property, interval graphs, and graph planarity using PQ-tree algorithrns - BOOTIt, LUEKER - 1976 |

1 |
TARJAN, An O(rnlogn)-time algorithm for the maximal subgraph problem
- CA, HAN, et al.
- 1993
(Show Context)
Citation Context ...oximation of a maximum-weight planar subgraph [14]. As a corollary of Theorem 1.1, we can construct a lexicographically maximum planar subgraph in O(rnlogn) time. The same result has been obtained in =-=[7]-=- with a different technique. A maximal planar subgraph of G can instead be constructed in O(n + rn) time [49] using the technique of 8.2. Transitive closure. In this section, we study the maintenance ... |

1 |
TAMASSlA, A19oritms for plane representations of acyclic digraphs Theoret
- unknown authors
- 1988
(Show Context)
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... |

1 |
graphs algorithms with SPQR-trees
- On-line
- 1990
(Show Context)
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... |

1 |
maintenance of triconnected components with SPQR-trees
- On-line
- 1996
(Show Context)
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... |

1 | FRaYSSEIX aN P. ROSENSTIEHL, A depth-first-search characterization of planarity - E - 1982 |

1 |
FREDEaICKSON, Data structures for on-line updating of minimum spanning trees with applications
- N
- 1985
(Show Context)
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 |
algoritms for shortest paths in planar graphs, with applications
- Fast
(Show Context)
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... |

1 |
data structures for dynamic 2-edge-connectivity and k smallest spanning
- Ambivalent
- 1991
(Show Context)
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... |

1 |
Fully dynamic algorithms for edge-connectivity problems
- unknown authors
- 1991
(Show Context)
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... |