### Citations

590 |
Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms,
- Booth, Lueker
- 1976
(Show Context)
Citation Context ...ique-separator graph of G from a clique path of G. There are numerous algorithms that recognize an interval graph and, if it is an interval graph, compute a clique path of the graph. Booth and Lueker =-=[BL76]-=- gave the first linear time algorithm and Corneil, Olariu, and Stewart [COS98] gave a much simpler algorithm, which computes a clique path and an interval representation of G. Both algorithms run in Θ... |

418 |
Graph classes: A survey ,
- Brandstadt, Le, et al.
- 1999
(Show Context)
Citation Context ...n the corresponding sections. Golumbic [Gol04] and McKee and McMorris [MM99] discuss these classes in the context of perfect graphs and intersection graphs, respectively. Brandstädt, Le, and Spinrad =-=[BLS99]-=- also survey these classes and Johnson [Joh85] discusses them with respect to the hardness of problems. We will use some terms from partially ordered set theory. Let F be a family of sets. For S ∈ F ,... |

306 |
Addendum: simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs,
- Tarjan, Yannakakis
- 1985
(Show Context)
Citation Context ...ty 2 with Ŝ = K. 7 Computing the clique-separator graph There are several algorithms that recognize a chordal graph and, if it is chordal, compute a clique tree of the graph. Tarjan and Yannakakis =-=[TY84]-=- gave an algorithm that recognizes chordal graphs and implicitly computes a clique tree and the maximal cliques of the graph. Lewis, Peyton, and Pothen [LPP89] gave an algorithm that computes the same... |

282 |
Incidence matrices and interval graphs.
- Fulkerson, Gross
- 1965
(Show Context)
Citation Context ...ly. A maximal clique corresponds to exactly one node of T and a minimal vertex separator may correspond to more than one edge of T . Furthermore, since a chordal graph G has at most n maximal cliques =-=[FG65]-=-, G has at most n− 1 minimal vertex separators. Interval graphs, proper interval graphs, and split graphs are subclasses of chordal graphs and will be defined in the corresponding sections. Golumbic [... |

239 | The NP-completeness column: an ongoing guide,
- Johnson
- 1987
(Show Context)
Citation Context ... and McKee and McMorris [MM99] discuss these classes in the context of perfect graphs and intersection graphs, respectively. Brandstädt, Le, and Spinrad [BLS99] also survey these classes and Johnson =-=[Joh85]-=- discusses them with respect to the hardness of problems. We will use some terms from partially ordered set theory. Let F be a family of sets. For S ∈ F , S is a minimal element of F if no S ′ ∈ F sat... |

170 | The intersection graphs of subtrees in trees are exactly the chordal graphs, - Gavril - 1974 |

152 | An introduction to chordal graphs and clique trees.
- Blair, Peyton
- 1993
(Show Context)
Citation Context ...: for any two maximal cliques K and K ′, the set K ∩ K ′ is contained in every maximal clique on the K–K ′ path in T [Bun74, Gav74, Wal78]. The clique tree is not necessarily unique. Blair and Peyton =-=[BP93]-=- discuss various properties of clique trees, including the following correspondence between the edges of T and the minimal vertex separators of G. (In [BP93], G is a connected chordal graph and then t... |

152 | On rigid circuit graphs, - Dirac - 1961 |

108 |
The Existential Graphs of
- Roberts
- 1973
(Show Context)
Citation Context ...to three nonadjacent vertices) is an interval graph but not a proper interval graph. In fact, an interval graph is a proper interval graph if and only if it has no induced subgraph isomorphic to K1,3 =-=[Rob69]-=-. An independent set of G is a set of pairwise nonadjacent vertices. The next theorem presents the properties of G when G is a proper interval graph. If G is not connected, the theorem can be applied ... |

97 | A characterization of rigid circuit graphs. - Buneman - 1974 |

67 | Power of natural semijoins. - Bernstein, Goodman - 1981 |

53 |
A.Pothen, A fast algorithm for reordering sparse matrices for parallel factorization
- Lewis
- 1989
(Show Context)
Citation Context ... tree of the graph. Tarjan and Yannakakis [TY84] gave an algorithm that recognizes chordal graphs and implicitly computes a clique tree and the maximal cliques of the graph. Lewis, Peyton, and Pothen =-=[LPP89]-=- gave an algorithm that computes the same explicitly, and this algorithm was simplified by Blair and Peyton [BP93]. Each of these algorithms runs in Θ(m + n) time. Given a chordal graph G, the followi... |

46 | The ultimate interval graph recognition algorithm,
- Corneil, Oleariu, et al.
- 1998
(Show Context)
Citation Context ...hms that recognize an interval graph and, if it is an interval graph, compute a clique path of the graph. Booth and Lueker [BL76] gave the first linear time algorithm and Corneil, Olariu, and Stewart =-=[COS98]-=- gave a much simpler algorithm, which computes a clique path and an interval representation of G. Both algorithms run in Θ(m + n) time. Given an interval representation of G, let I(v) be the interval ... |

33 | Counting clique trees and computing perfect elimination schemes - HO, LEE - 1989 |

30 | A fully dynamic algorithm for recognizing and representing proper interval graphs,
- Hell, Shamir, et al.
- 2002
(Show Context)
Citation Context ...val graph have been used to simplify the algorithm in [Iba07] so that it recognizes proper interval graphs in O(logn) time per update or query [Iba08]. This algorithm is simpler than the algorithm in =-=[HSS01]-=- and it matches its running time for edge updates and connectivity queries. In comparison, algorithms that recognize an interval graph or proper interval graph from scratch run in O(m + n) time, where... |

29 |
Algorithmic Graph Theory and Perfect
- Golumbic
- 2004
(Show Context)
Citation Context ...], G has at most n− 1 minimal vertex separators. Interval graphs, proper interval graphs, and split graphs are subclasses of chordal graphs and will be defined in the corresponding sections. Golumbic =-=[Gol04]-=- and McKee and McMorris [MM99] discuss these classes in the context of perfect graphs and intersection graphs, respectively. Brandstädt, Le, and Spinrad [BLS99] also survey these classes and Johnson ... |

28 |
Intersection graphs of paths in a tree,
- Monma, Wei
- 1986
(Show Context)
Citation Context ...have {interval graphs} ⊆ {RDV graphs}. Johnson [Joh85] refers to RDV graphs as directed path graphs in his discussion of 19 “algorithmically significant classes of intersection graphs”. Monma and Wei =-=[MW86]-=- present a unified framework for studying intersection graphs of families of paths in a tree and they characterize RDV graphs as well as larger subclasses of chordal graphs. Figure 13 shows (a) a chor... |

26 | Representations of Chordal Graphs as Subtrees of a Tree - Walter - 1978 |

20 | Chordal graphs and their clique graph
- Galinier, Habib, et al.
- 1995
(Show Context)
Citation Context ...as more than exponentially many clique trees. The clique-separator graph is different from the clique graph in [Shi88], the weighted clique intersection graph in [BP93, BG81], and the clique graph in =-=[GHP95]-=-. The graph in [Shi88] is the intersection graph of the set of maximal cliques of a chordal graph G, i.e., the nodes are the maximal cliques and two nodes are adjacent exactly when the corresponding c... |

17 |
On the tree representation of chordal graphs
- Shibata
- 1988
(Show Context)
Citation Context ...rtices has exactly one clique tree, which is isomorphic to a clique tree of K1,n, which has more than exponentially many clique trees. The clique-separator graph is different from the clique graph in =-=[Shi88]-=-, the weighted clique intersection graph in [BP93, BG81], and the clique graph in [GHP95]. The graph in [Shi88] is the intersection graph of the set of maximal cliques of a chordal graph G, i.e., the ... |

11 | Zero patterns, chordal graphs and matrix completions - Lundquist - 1990 |

6 | A fully dynamic graph algorithm for recognizing interval graphs
- Ibarra
- 2007
(Show Context)
Citation Context ...r the minimal vertex separators have this property. The structural properties of G when G is an interval graph have been used to develop the first dynamic graph algorithm to recognize interval graphs =-=[Iba07]-=-. A dynamic graph algorithm maintains a solution to a graph problem as the graph undergoes a series of small changes, such as single edge insertions or edge deletions. For every change, the algorithm ... |

3 | Laminar structure of ptolemaic graphs and its applications
- Uehara, Uno
- 2005
(Show Context)
Citation Context ...h’s nodes are the maximal cliques and the minimal vertex separators of G and it has both arcs and edges. The clique-separator graph is also different from the clique laminar tree for ptolemaic graphs =-=[UU05]-=-, which are the intersection of chordal graphs and distance hereditary graphs. The clique laminar tree is a directed tree where the nodes are the sets of nonempty intersections of the maximal cliques ... |

2 | A simple dynamic graph algorithm for recognizing proper interval graphs. Manuscript. Preliminary version available for download at http://facweb.cs.depaul.edu/ibarra/research.htm - Ibarra - 2007 |

1 |
Representing chordal graphs on K1,n
- McMorris, Shier
- 1983
(Show Context)
Citation Context ...ree intervals. A chordal graph is the intersection graph of a family of subtrees of a tree [Bun74, Gav74, Wal78] and a split graph is the intersection graph of a family of distinct subtrees of a star =-=[MS83]-=-, where each subtree is considered to be a set of vertices. In Section 4, we showed that the clique-separator graphs of interval graphs have structural properties not shared by the clique-separator gr... |