## Diameter determination on restricted graph families (1998)

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Venue: | In WG 98. 24th International Workshop on Graph Theoretic Concept in Computer Science |

Citations: | 14 - 4 self |

### BibTeX

@INPROCEEDINGS{Corneil98diameterdetermination,

author = {Derek G. Corneil and Michel Habib and Christophe Paul},

title = {Diameter determination on restricted graph families},

booktitle = {In WG 98. 24th International Workshop on Graph Theoretic Concept in Computer Science},

year = {1998},

pages = {192--202},

publisher = {Springer}

}

### Years of Citing Articles

### OpenURL

### Abstract

### Citations

1186 | Algorithmic graph theory and perfect graphs - Golumbic - 1980 |

840 | Matrix multiplication via arithmetic progressions - Coppersmith, Winograd - 1987 |

309 |
Algorithmic aspects of vertex elimination on graphs
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- 1976
(Show Context)
Citation Context ...A graph is AT-free if it does not contain an AT. The algorithm that we present involves two sweeps of the well-known Lexicographic Breadth First Search (Lex-BFS) introduced by Rose, Tarjan and Lueker =-=[11]-=- (see algorithm 1) for the recognition of chordal graphs. An example of a Lex-BFS sweep is presented in figure 7. It is somewhat surprising that Lex-BFS seems to play a fundamental role for both chord... |

251 | Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs - Tarjan, Yannakakis - 1984 |

139 | 1962 “Representation of a finite graph by a set of intervals on a real line”. Fundamwta Mathematicne 51: 45-64 - Boland |

133 | The Intersection Graphs of Subtrees in a Tree are Exactly the Chordal Graphs - Gavril - 1974 |

68 | Fast estimation of diameter and shortest paths (without matrix multiplication - Aingworth, Motwani - 1996 |

55 | Asteroidal triple-free graphs
- Corneil, Olariu, et al.
- 1997
(Show Context)
Citation Context ... in figure 7. It is somewhat surprising that Lex-BFS seems to play a fundamental role for both chordal and AT-free graphs, two families that exhibit very little structural similarity (see for example =-=[3, 4, 8, 10, 11]-=-). Furthermore Dragan et al [6] and Dragan [5] have shown that the eccentricity of the vertex visited last in a Lex-BFS is within 1 of the diameter for chordal and AT-free graphs respectively. Algorit... |

42 |
Characterizations of strongly chordal graphs
- Farber
- 1983
(Show Context)
Citation Context ...v). A graph is chordal iff there is no chordless cycle of length more than 3. A chord xixj in a cycle C = (x1; x2; : : : ; x2n) is an odd chord if, in C, d(xi; xj) is odd. A graph is strongly chordal =-=[7]-=- if it is chordal and each cycle of even length at least 6 has an odd chord. Three vertices u; v; w are an asteroidal triple (AT) if \Lambda The proofs on chordal graphs are available in french on the... |

41 | On the all-pairs-shortest-path problem - Seidel - 1992 |

36 | Characterizations of strongly chordal graphs, Discrete Math - Farber - 1983 |

26 | Linear time algorithms for dominating pairs in asteroidal triple-free graphs - Corneil, Olariu, et al. - 1999 |

24 |
The algorithmic use of hypertree structure and maximum neighbourhood orderings
- Brandstädt, Chepoi, et al.
- 1998
(Show Context)
Citation Context ... works optimally for strongly chordal graphs. The only other linear time algorithm for calculating the diameter of strongly chordal graphs is somewhat complicated and much more difficult to implement =-=[1]-=-. Before presenting these results, we show that it is unlikely that the diameter 2sproblem on either chordal or AT-free graphs can be solved in quadratic time. To do this we introduce the disjoint set... |

22 | A recognition algorithm for the intersection graphs of directed paths in directed trees - Gavril - 1975 |

21 | On the semi-perfect elimination - Jamison, Olariu - 1988 |

18 | Dominating cliques in distance-hereditary graphs - Dragan - 1994 |

17 | On distances in benzenoid systems - Chepoi - 1996 |

16 | Minimax location of a facility in an undirected tree graph - Handler - 1973 |

13 |
The intersection graphs of a path in a tree are exactly the chordal graphs
- Gavril
- 1974
(Show Context)
Citation Context ... in figure 7. It is somewhat surprising that Lex-BFS seems to play a fundamental role for both chordal and AT-free graphs, two families that exhibit very little structural similarity (see for example =-=[3, 4, 8, 10, 11]-=-). Furthermore Dragan et al [6] and Dragan [5] have shown that the eccentricity of the vertex visited last in a Lex-BFS is within 1 of the diameter for chordal and AT-free graphs respectively. Algorit... |

9 | A simple linear-time algorithm for computing the center of an interval graph - Olariu - 1990 |

8 | A characterization of rigid circuit graphs. Discrete Math - Buneman - 1974 |

7 | Lexbfs-orderings and powers of graphs - Dragan, Nicolai, et al. - 1997 |

7 | Computation of the center and diameter of outerplanar graphs - Farley, Proskurowski - 1980 |

7 | Representation of a nite graph by a set of intervals on the real line - Boland |

6 | Lexbfsorderings of distance-hereditary graphs with application to the diametral pair problem - Dragan, Nicolai |

6 | and Andrzej Proskurowski. Computation of the center and diameter of outerplanar graphs - Farley - 1980 |

5 | Unified all-pairs shortest path algorithms in the chordal hierarchy - Han, Sekharan, et al. - 1997 |

4 |
Parcours en largeur lexicographique: un algorithme de partitionnement, application aux graphes et g'en'eralisation
- Paul
- 1998
(Show Context)
Citation Context ... in figure 7. It is somewhat surprising that Lex-BFS seems to play a fundamental role for both chordal and AT-free graphs, two families that exhibit very little structural similarity (see for example =-=[3, 4, 8, 10, 11]-=-). Furthermore Dragan et al [6] and Dragan [5] have shown that the eccentricity of the vertex visited last in a Lex-BFS is within 1 of the diameter for chordal and AT-free graphs respectively. Algorit... |

4 | Optimal (parallel) algorithms for the all-to-all vertex distances problem for certain graph classes - Dahlhaus - 1993 |

3 | Almost diameter of a house-holefree graph in linear time via lexbfs. Discrete Applied Mathematics - Dragan - 1999 |

1 |
Disjoint sets problem. private communication
- Chepoi
- 1996
(Show Context)
Citation Context ...exist i and j such that Si "Sj = ;. To our knowledge, there is no algorithm for this problem with a better running time than the order of Boolean Matrix Multiplication (BMM). As pointed out by Chepoi =-=[2]-=-, a fast algorithm (i.e. in quadratic time or better) for determining whether a split graph (and thus a chordal graph) has diameter 2 or 3 would imply a fast algorithm (i.e. faster than BMM) for the D... |

1 |
Almost diameter in hhd-free graphs in linear time via lexbfs
- Dragan
- 1997
(Show Context)
Citation Context ...o play a fundamental role for both chordal and AT-free graphs, two families that exhibit very little structural similarity (see for example [3, 4, 8, 10, 11]). Furthermore Dragan et al [6] and Dragan =-=[5]-=- have shown that the eccentricity of the vertex visited last in a Lex-BFS is within 1 of the diameter for chordal and AT-free graphs respectively. Algorithm 1: Lexicographic Breadth First Search (Lex-... |

1 |
Lex-bfs orderings and powers of graphs
- Dragan, Nicolai, et al.
- 1997
(Show Context)
Citation Context ...Lex-BFS seems to play a fundamental role for both chordal and AT-free graphs, two families that exhibit very little structural similarity (see for example [3, 4, 8, 10, 11]). Furthermore Dragan et al =-=[6]-=- and Dragan [5] have shown that the eccentricity of the vertex visited last in a Lex-BFS is within 1 of the diameter for chordal and AT-free graphs respectively. Algorithm 1: Lexicographic Breadth Fir... |

1 |
Couplages maximaux et diam`etre de graphes
- Lesk
- 1984
(Show Context)
Citation Context ...ssical and well-known problem. For arbitrary graphs, as well as for various restricted graph families, the current fastest algorithm for this problem achieves the time bound of O(nm) (see for example =-=[9]-=-) which is too slow to be practical for very large graphs. In this paper, we study the problem of determining a vertex of high eccentricity for chordal and AT-free graphs. The eccentricity of a vertex... |

1 | Disjoint sets problem, unpublished - Chepoi, Dragan - 1992 |

1 | Uni ed all-pairs shortest path algorithms in the chordal hierarchy - Han, Sekharan, et al. - 1997 |