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34
Minimal triangulations of graphs: A survey
 DISCRETE MATHEMATICS
"... 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 ..."
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Cited by 37 (3 self)
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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.
A Fully Dynamic Algorithm for Recognizing and Representing Proper Interval Graphs
 SIAM J. COMPUT
, 1999
"... In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to ..."
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Cited by 30 (1 self)
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In this paper we study the problem of recognizing and representing dynamically changing proper interval graphs. The input to the problem consists of a series of modifications to be performed on a graph, where a modification can be a deletion or an addition of a vertex or an edge. The objective is to maintain a representation of the graph as long as it remains a proper interval graph, and to detect when it ceases to be so. The representation should enable one to efficiently construct a realization of the graph by an inclusionfree family of intervals. This problem has important applications in physical mapping of DNA. We give a nearoptimal fully dynamic algorithm for this problem. It operates in time O(log n) per edge insertion or deletion. We prove a close lower bound of\Omega\Gamma/24 n=(log log n + log b)) amortized time per operation, in the cell probe model with wordsize b. We also construct optimal incremental and decremental algorithms for the problem, which handle each edge operation in O(1) time.
A Fully Dynamic Algorithm for Modular Decomposition and Recognition of Cographs
 Discrete Appl. Math
, 2004
"... The problem of dynamically recognizing a graph property calls for efficiently deciding if an input graph satisfies the property under repeated modifications to its set of vertices and edges. The input to the problem consists of a series of modifications to be performed on the graph. The objective is ..."
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Cited by 14 (0 self)
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The problem of dynamically recognizing a graph property calls for efficiently deciding if an input graph satisfies the property under repeated modifications to its set of vertices and edges. The input to the problem consists of a series of modifications to be performed on the graph. The objective is to maintain a representation of the graph as long as the property holds, and to detect when it ceases to hold.
A vertex incremental approach for maintaining chordality
 Discrete Mathematics
, 2006
"... For a chordal graph G = (V, E), we study the problem of whether a new vertex u � ∈ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maxim ..."
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Cited by 11 (5 self)
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For a chordal graph G = (V, E), we study the problem of whether a new vertex u � ∈ V and a given set of edges between u and vertices in V can be added to G so that the resulting graph remains chordal. We show how to resolve this efficiently, and at the same time, if the answer is no, specify a maximal subset of the proposed edges that can be added along with u, or conversely, a minimal set of extra edges that can be added in addition to the given set, so that the resulting graph is chordal. In order to do this, we give a new characterization of chordal graphs and, for each potential new edge uv, a characterization of the set of edges incident to u that also must be added to G along with uv. We propose a data structure that can compute and add each such set in O(n) time. Based on these results, we present an algorithm that computes both a minimal triangulation and a maximal chordal subgraph of an arbitrary input graph in O(nm) time, using a totally new vertex incremental approach. In contrast to previous algorithms, our process is online in that each new vertex is added without reconsidering any choice made at previous steps, and without requiring any knowledge of the vertices that might be added subsequently. 1
Fully Dynamic Algorithm for Recognition and Modular Decomposition of Permutation Graphs
 ALGORITHMICA
, 2009
"... This paper considers the problem of maintaining a compact representation (O(n) space) of permutation graphs under vertex and edge modifications (insertion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposit ..."
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Cited by 8 (3 self)
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This paper considers the problem of maintaining a compact representation (O(n) space) of permutation graphs under vertex and edge modifications (insertion or deletion). That representation allows us to answer adjacency queries in O(1) time. The approach is based on a fully dynamic modular decomposition algorithm for permutation graphs that works in O(n) time per edge and vertex modification. We thereby obtain a fully dynamic algorithm for the recognition of permutation graphs.
Adding an edge in a cograph
 In 31st International Workshop on GraphTheoretic Concepts in Computer Science, WG 2005, LNCS Proceedings
, 2005
"... Abstract. In this paper, we establish structural properties of cographs which enable us to present an algorithm which, for a cograph G and anonedgexy (i.e., two nonadjacent vertices x and y) ofG, finds the minimum number of edges that need to be added to the edge set of G such that the resulting g ..."
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Cited by 6 (0 self)
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Abstract. In this paper, we establish structural properties of cographs which enable us to present an algorithm which, for a cograph G and anonedgexy (i.e., two nonadjacent vertices x and y) ofG, finds the minimum number of edges that need to be added to the edge set of G such that the resulting graph is a cograph and contains the edge xy. The motivation for this problem comes from algorithms for the dynamic recognition and online maintenance of graphs; the proposed algorithm could be a suitable addition to the algorithm of Shamir and Sharan [13] for the online maintenance of cographs. The proposed algorithm runs in time linear in the size of the input graph and requires linear space.
Dynamic distance hereditary graphs using split decomposition
 In Algorithms and Computation, 18th International Symposium (ISAAC
, 2007
"... Abstract. The problem of maintaining a representation of a dynamic graph as long as a certain property is satisfied has recently been considered for a number of properties. This paper presents an optimal algorithm for this problem on vertexdynamic connected distance hereditary graphs: both vertex i ..."
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Cited by 6 (4 self)
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Abstract. The problem of maintaining a representation of a dynamic graph as long as a certain property is satisfied has recently been considered for a number of properties. This paper presents an optimal algorithm for this problem on vertexdynamic connected distance hereditary graphs: both vertex insertion and deletion have complexity O(d), where d is the degree of the vertex involved in the modification. Our vertexdynamic algorithm is competitive with the existing linear time recognition algorithms of distance hereditary graphs, and is also simpler. Besides, we get a constant time edgedynamic recognition algorithm. To achieve this, we revisit the split decomposition by introducing graphlabelled trees. Doing so, we are also able to derive an intersection model for distance hereditary graphs, which answers an open problem. 1
Fullydynamic recognition algorithm and certificate for directed cographs
 Proc. 30th Int’l Workshop on GraphTheoretic Concepts in Computer Science (WG’04), LNCS3353
, 2004
"... Abstract. This paper presents an optimal fullydynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) timewhered is the number of arcs involved in the operat ..."
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Cited by 5 (4 self)
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Abstract. This paper presents an optimal fullydynamic recognition algorithm for directed cographs. Given the modular decomposition tree of a directed cograph G, the algorithm supports arc and vertex modification (insertion or deletion) in O(d) timewhered is the number of arcs involved in the operation. Moreover, if the modified graph remains a directed cograph, the modular tree decomposition is updated; otherwise, a certificate is returned within the same complexity. 1
Towards Improving Phylogeny Reconstruction with CombinatorialBased Constraints on an Underlying Family of Graphs
, 2002
"... We address the issue of improving phylogenetic data by modifying a dissimilarity matrix so that it will be closer to an additive matrix (the matrix corresponding to a phylogeny), in the case where the thresholds are known to be too low. Our approach uses the thresholds defined by the input matrix t ..."
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Cited by 4 (2 self)
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We address the issue of improving phylogenetic data by modifying a dissimilarity matrix so that it will be closer to an additive matrix (the matrix corresponding to a phylogeny), in the case where the thresholds are known to be too low. Our approach uses the thresholds defined by the input matrix to define a family of pairwise inclusive undirected graphs. We run an experimental analysis on a previous algorithm, which leads us to a combinatorial study of all the successions of subgraphs on four vertices which an additive matrix can span, thereby extracting some interesting rules on the additive family of graphs.