Asteroidal Triple-Free Graphs

Corneil, Derek G.; Olariu, Stephan; Stewart, Lorna

doi:10.1137/s0895480193250125

Cited by 137 publications

(67 citation statements)

References 13 publications

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“…From this result, it seems a natural question to approach this problem to other subclasses of AT-free families [3], such as permutation and trapezoid graphs, both of them being also perfect and intersection graphs, as interval graphs are.…”

Section: Definition 17mentioning

confidence: 99%

On the Steiner, geodetic and hull numbers of graphs

Hernando

Jiang

Mora

et al. 2005

Discrete Mathematics

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Given a graph G and a subset W ⊆ V (G), a Steiner W -tree is a tree of minimum order that contains all of W. Let S(W ) denote the set of all vertices in G that lie on some Steiner W-tree; we call S(W ) the Steiner interval of W. If S(W ) = V (G), then we call W a Steiner set of G. The minimum order of a Steiner set of G is called the Steiner number of G.Giventhe set of all vertices in G lying on some u-v geodesic, and let J [u, v] denote the set of all vertices in G lying on some induced u-v path. Given a set S ⊆ V (G), let I [S] = u,v∈S I [u, v], and let J [S] = u,v∈S J [u, v]. We call I [S] the geodetic closure of S and J [S] the monophonic closure of S. If I [S] = V (G), then S is called a geodetic set of G. If J [S] = V (G), then S is called a monophonic set of G. The minimum order of a geodetic set in G is named the geodetic number of G.In this paper, we explore the relationships both between Steiner sets and geodetic sets and between Steiner sets and monophonic sets. 140 C. Hernando et al. / Discrete Mathematics 293 (2005) 139 -154 and the geodetic number, and address the following questions: in a graph G when must every Steiner set also be geodetic and when must every Steiner set also be monophonic. In particular, amongothers we show that every Steiner set in a connected graph G must also be monophonic, and that every Steiner set in a connected interval graph H must be geodetic.

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Section: Definition 17mentioning

confidence: 99%

On the Steiner, geodetic and hull numbers of graphs

Hernando

Jiang

Mora

et al. 2005

Discrete Mathematics

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“…Lekkerkerker and Boland [63] showed that a graph is an interval graph if and only if it is chordal and AT-free. The only-if part of the following characterization of AT-free graphs through their minimal triangulations was proved by Möhring [68], and the if part was proved independently by Corneil et al [27], and by Parra and Scheffler [74]. Theorem 7.1 (Corneil et al [27], Möhring [68], Parra and Scheffler [74]…”

Section: Minimal Triangulation Of Restricted Graph Classesmentioning

confidence: 93%

Minimal triangulations of graphs: A survey

Heggernes

2006

Discrete Mathematics

156

112

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Any given graph can be embedded in a chordal graph by adding edges, and the resulting chordal graph is called a triangulation of the input graph. In this paper we study minimal triangulations, which are the result of adding an inclusion minimal set of edges to produce a triangulation. This topic was first studied from the standpoint of sparse matrices and vertex elimination in graphs. Today we know that minimal triangulations are closely related to minimal separators of the input graph. Since the first papers presenting minimal triangulation algorithms appeared in 1976, several characterizations of minimal triangulations have been proved, and a variety of algorithms exist for computing minimal triangulations of both general and restricted graph classes. This survey presents and ties together these results in a unified modern notation, keeping an emphasis on the algorithms.

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“…The algorithm starts with a preprocessing in which it computes A 2 in time O(n 2.376 ) by matrix multiplication [14], where A is the adjacency matrix of G for which A(i, j) = 1 if i ≠ j and {v i , v j } ∈ E, and A(i, j) = 0 otherwise. Consequently, during the dynamic programming part of the algorithm, the number of common neighbors of two vertices v i and v j can be computed in constant time.…”

Section: Interval Graphsmentioning

confidence: 99%

Degree-preserving trees

et al. 2000

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We consider the degree‐preserving spanning tree (DPST) problem: Given a connected graph G, find a spanning tree T of G such that as many vertices of T as possible have the same degree in T as in G. This problem is a graph‐theoretical translation of a problem arising in the system‐theoretical context of identifiability in networks, a concept which has applications in, for example, water distribution networks and electrical networks. We show that the DPST problem is NP‐complete, even when restricted to split graphs or bipartite planar graphs, but that it can be solved in polynomial time for graphs with a bounded asteroidal number and for graphs with a bounded treewidth. For the class of interval graphs, we give a linear time algorithm. For the class of cocomparability graphs, we give an O(n4) algorithm. Furthermore, we present linear time approximation algorithms for planar graphs of a worst‐case performance ratio of 1 − ϵ for every ϵ > 0. © 2000 John Wiley & Sons, Inc.

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Asteroidal Triple-Free Graphs

Cited by 137 publications

References 13 publications

On the Steiner, geodetic and hull numbers of graphs

On the Steiner, geodetic and hull numbers of graphs

Minimal triangulations of graphs: A survey

Degree-preserving trees

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