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22
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co ..."
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Cited by 188 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
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 55 (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...
Efficient parallel algorithms for chordal graphs
"... We give the first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadthfirst search tree and a depthfirst search tree of a chordal graph, recognizing ..."
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Cited by 26 (0 self)
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We give the first efficient parallel algorithms for recognizing chordal graphs, finding a maximum clique and a maximum independent set in a chordal graph, finding an optimal coloring of a chordal graph, finding a breadthfirst search tree and a depthfirst search tree of a chordal graph, recognizing interval graphs, and testing interval graphs for isomorphism. The key to our results is an efficient parallel algorithm for finding a perfect elimination ordering.
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 25 (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...
A simple test for interval graphs
 Proc. 18th Int. Workshop (WG '92), GraphTheoretic Concepts in Computer Science
, 1992
"... An interval graph is the intersection graph of a collection of intervals. Interval graphs are a special class of chordal graphs. This class of graphs has a wide range of applications. Several linear time algorithms have been designed to recognize interval graphs. Booth & Lueker first used PQtrees t ..."
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Cited by 19 (2 self)
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An interval graph is the intersection graph of a collection of intervals. Interval graphs are a special class of chordal graphs. This class of graphs has a wide range of applications. Several linear time algorithms have been designed to recognize interval graphs. Booth & Lueker first used PQtrees to recognize interval graphs in linear time. However, the data manipulation of PQtrees is rather involved and the complexity analysis is also quite tricky. Korte and Möhring simplified the operations on a PQtree using an incremental algorithm. Hsu and Ma gave a simpler decomposition algorithm without using PQtrees. All of these algorithms rely on the following fact: a graph is an interval graph iff there exists a linear order of its maximal cliques such that for each vertex v, all maximal cliques containing v are consecutive. Thus, the precomputation of all maximal cliques is required for these algorithms. Based on graph decomposition, we give a much simpler recognition algorithm in this paper which directly places the intervals without precomputing all maximal cliques. A linear time isomorphism algorithm can be easily derived as a byproduct. Another advantage of our approach is that it can be used to develop an O(nlog n) online recognition algorithm for interval graphs. 1.
An Heuristic For Graph Symmetry Detection
 Proc. of Graph Drawing 99, Lecture Notes in Computer Science 1731:276285
, 1999
"... . We give a short introduction to an heuristic to find automorphisms in a graph such as axial, central or rotational symmetries. Using technics of factorial analysis, we embed the graph in an Euclidean space and try to detect and interpret the geometric symmetries of of the embedded graph. 1. Introd ..."
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Cited by 6 (1 self)
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. We give a short introduction to an heuristic to find automorphisms in a graph such as axial, central or rotational symmetries. Using technics of factorial analysis, we embed the graph in an Euclidean space and try to detect and interpret the geometric symmetries of of the embedded graph. 1. Introduction Testing whether a graph has any axial (rotational, central, respectively) symmetry is a NPcomplete problem [9]. Some restrictions (central symmetry with exactly one fixed vertex and no fixed edge) are polynomialy equivalent to the graph isomorphism test. Notice that this latter problem is not known to be either polynomial or NPcomplete in general. But several heuristics are known (e.g. [3]) and several restrictions leads to efficient algorithms: linear time isomorphism test for planar graphs [6] and interval graphs [8], polynomial time isomorphism test for fixed genus [10, 5], kcontractible graphs [12] and pairwise kseparable graphs [11], linear axial symmetry detection for plana...
Capturing polynomial time on interval graphs
 in Proceedings of the 25th IEEE Symposium on Logic in Computer Science, 2010, this volume
"... The present paper proves a characterization of all polynomialtime computable queries on the class of interval graphs by sentences of fixedpoint logic with counting. The result is one of the first establishing the capturing of polynomial time on a graph class which is defined by forbidden induced s ..."
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Cited by 5 (1 self)
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The present paper proves a characterization of all polynomialtime computable queries on the class of interval graphs by sentences of fixedpoint logic with counting. The result is one of the first establishing the capturing of polynomial time on a graph class which is defined by forbidden induced subgraphs. More precisely, it is shown that on the class of unordered interval graphs, any query is polynomialtime computable if and only if it is definable in fixedpoint logic with counting. Furthermore, it is shown that fixedpoint logic is not expressive enough to capture polynomial time on the classes of chordal graphs or incomparability graphs. 1
On Linear and Circular Structure of (claw, net)Free Graphs
, 2003
"... We prove that every (claw, net)free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present alS[SE timealen##ES which, for a given (claw, net)free graph, finds either a dominating pair or an induceddoubl dominatingcycln We show aln how one can uses ..."
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Cited by 4 (3 self)
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We prove that every (claw, net)free graph contains an induced doubly dominating cycle or a dominating pair. Moreover, using LexBFS we present alS[SE timealen##ES which, for a given (claw, net)free graph, finds either a dominating pair or an induceddoubl dominatingcycln We show aln how one can usestructural properties of (claw, net)free graphs tosolI efficiently the domination, independent domination, and independent set problems on these graphs.
Computing a Dominating Pair in an Asteroidal Triplefree Graph in Linear Time
 in Algorithms and Data Structures WADS '95, Lecture
, 1998
"... 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 neighborhood of the third. A graph is asteroidal triplefree (ATfree, for short) if it contains no asteroidal triple. The motivation for this work is prov ..."
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Cited by 3 (2 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 neighborhood 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 can be implemented to run in time linear in the size of the input, whereas the best algorithm previously known for this problem has complexity O(jV j 3 ) for input...
A Fast Parallel Algorithm to Recognize P 4 sparse Graphs
 Discrete Appl. Math
"... A number of problems in computational semantics, groupbased collaboration, automated theorem proving, networking, scheduling, and cluster analysis suggested the study of graphs featuring certain "local density" characteristics. Typically, the notion of local density is equated with the absence of c ..."
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Cited by 3 (1 self)
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A number of problems in computational semantics, groupbased collaboration, automated theorem proving, networking, scheduling, and cluster analysis suggested the study of graphs featuring certain "local density" characteristics. Typically, the notion of local density is equated with the absence of chordless paths of length three or more. Recently, a new metric for local density has been proposed, allowing a number of such induced paths to occur. More precisely, a graph G is called P4sparse if no set of five vertices in G induces more than one chordless path of length three. P4sparse graphs generalize the wellknown class of cographs corresponding to a more stringent local density metric. One remarkable feature of P4sparse graphs is that they admit a tree representation unique up to isomorphism. In this work we present a parallel algorithm to recognize P4sparse graphs and show how the data structures returned by the recognition algorithm can be used to construct the corresponding tr...