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178
Complexity and Algorithms for Reasoning About Time: A GraphTheoretic Approach
, 1992
"... Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence ..."
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Cited by 98 (11 self)
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Temporal events are regarded here as intervals on a time line. This paper deals with problems in reasoning about such intervals when the precise topological relationship between them is unknown or only partially specified. This work unifies notions of interval algebras in artificial intelligence with those of interval orders and interval graphs in combinatorics. The satisfiability, minimal labeling, all solutions and all realizations problems are considered for temporal (interval) data. Several versions are investigated by restricting the possible interval relationships yielding different complexity results. We show that even when the temporal data comprises of subsets of relations based on intersection and precedence only, the satisfiability question is NPcomplete. On the positive side, we give efficient algorithms for several restrictions of the problem. In the process, the interval graph sandwich problem is introduced, and is shown to be NPcomplete. This problem is als...
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"... A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (M ..."
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Cited by 81 (5 self)
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A rigid interval graph is an interval graph which has only one clique tree. In 2009, Panda and Das show that all connected unit interval graphs are rigid interval graphs. Generalizing the two classic graph search algorithms, Lexicographic BreadthFirst Search (LBFS) and Maximum Cardinality Search (MCS), Corneil and Krueger propose in 2008 the socalled Maximal Neighborhood Search (MNS) and show that one sweep of MNS is enough to recognize chordal graphs. We develop the MNS properties of rigid interval graphs and characterize this graph class in several different ways. This allows us obtain several linear time multisweep MNS algorithms for recognizing rigid interval graphs and unit interval graphs, generalizing a corresponding 3sweep LBFS algorithm for unit interval graph recognition designed by Corneil in 2004. For unit interval graphs, we even present a new linear time 2sweep MNS certifying recognition algorithm. Submitted:
Four Strikes against Physical Mapping of DNA
 JOURNAL OF COMPUTATIONAL BIOLOGY
, 1993
"... Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete ..."
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Cited by 67 (8 self)
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Physical Mapping is a central problem in molecular biology ... and the human genome project. The problem is to reconstruct the relative position of fragments of DNA along the genome from information on their pairwise overlaps. We show that four simplified models of the problem lead to NPcomplete decision problems: Colored unit interval graph completion, the maximum interval (or unit interval) subgraph, the pathwidth of a bipartite graph, and the kconsecutive ones problem for k >= 2. These models have been chosen to reflect various features typical in biological data, including false negative and positive errors, small width of the map and chimericism.
Asteroidal TripleFree Graphs
, 1997
"... . An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in ..."
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Cited by 61 (10 self)
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. An independent set of three vertices such that each pair is joined by a path that avoids the neighborhood of the third is called an asteroidal triple. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triples. The motivation for this investigation was provided, in part, by the fact that the asteroidal triplefree graphs provide a common generalization of interval, permutation, trapezoid, and cocomparability graphs. The main contribution of this work is to investigate and reveal fundamental structural properties of ATfree graphs. Specifically, we show that every connected ATfree graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. We then provide characterizations of ATfree graphs in terms of dominating pairs and minimal triangulations. Subsequently, we state and prove a decomposition theorem for ATfree graphs. An assortment of other properties of ATfree graphs is also p...
Tractability of Parameterized Completion Problems on Chordal, Strongly Chordal and Proper Interval Graphs
, 1994
"... We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k ..."
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Cited by 58 (5 self)
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We study the parameterized complexity of three NPhard graph completion problems. The MINIMUM FILLIN problem is to decide if a graph can be triangulated by adding at most k edges. We develop O(c m) and O(k mn + f(k)) algorithms for this problem on a graph with n vertices and m edges. Here f(k) is exponential in k and the constants hidden by the bigO notation are small and do not depend on k. In particular, this implies that the problem is fixedparameter tractable (FPT). The PROPER
On the Treewidth and Pathwidth of Permutation Graphs
, 1992
"... In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[r] is a permutation graph with treewidth k, then we show that the pathwidth of G[r] is at most 2k, and we give an algo rithm which constructs a pathdecomposition with width at most 2k in time ..."
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Cited by 51 (12 self)
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In this paper we discuss the problem of finding the treewidth and pathwidth of permutation graphs. If G[r] is a permutation graph with treewidth k, then we show that the pathwidth of G[r] is at most 2k, and we give an algo rithm which constructs a pathdecomposition with width at most 2k in time O(nk). We assume that the permutation r is given. For permutation graphs of which the treewidth is bounded by some constant, this result, together with previous results ([9], [15]), shows that the treewidth and pathwidth can be computed in linear time.
Certifying algorithms for recognizing interval graphs and permutation graphs
 SIAM J. COMPUT
, 2006
"... A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give lineartime certifying algorithms for recognition o ..."
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Cited by 44 (8 self)
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A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give lineartime certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(V) time.
Algorithms for Square Roots of Graphs
 SIAM Journal on Discrete Mathematics
, 1991
"... The nth power (n 1) of a graph G = (V; E), written G n , is defined to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an nth root G if G n = H . For the case of n = 2, ..."
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Cited by 40 (0 self)
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The nth power (n 1) of a graph G = (V; E), written G n , is defined to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an nth root G if G n = H . For the case of n = 2, we say that G 2 is the square of G and G is the square root of G 2 . Here we give a linear time algorithm for finding the tree square roots of a given graph and a linear time algorithm for finding the square roots of planar graphs. We also give a polynomial time algorithm for finding the square roots of subdivision graphs, which is equivalent to the problem of the inversion of total graphs. Further, we give a linear time algorithm for finding a Hamiltonian cycle in a cubic graph, and we prove the NPcompleteness of finding the maximum cliques in powers of graphs and the chordality of powers of trees. Keywords: Square graphs, power graphs, tree square, planar square graphs. 1 Introduct...
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 38 (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. 1 Introduction and
Linear Time Algorithms for Dominating Pairs in Asteroidal Triplefree Graphs
 SIAM J. Comput
, 1997
"... An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighbourhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is pro ..."
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Cited by 32 (7 self)
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An independent set of three of vertices is called an asteroidal triple if between each pair in the triple there exists a path that avoids the neighbourhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is provided, in part, by the fact that ATfree graphs offer a common generalization of interval, permutation, trapezoid, and cocomparability graphs. Previously, the authors have given an existential proof of the fact that every connected ATfree graph contains a dominating pair, that is, a pair of vertices such that every path joining them is a dominating set in the graph. The main contribution of this paper is a constructive proof of the existence of dominating pairs in connected ATfree graphs. The resulting simple algorithm, based on the wellknown Lexicographic BreadthFirst Search, can be implemented to run in time linear in the size of the input, whereas the best algorithm previousl...