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Modular Decomposition and Transitive Orientation
, 1999
"... A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular ..."
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Cited by 90 (14 self)
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A module of an undirected graph is a set X of nodes such for each node x not in X, either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linearspace representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We give O(n +m) algorithms for modular decomposition and transitive orientation, where n and m are the number of vertices and edges of the graph. This gives linear time bounds for recognizing permutation graphs, maximum clique and minimum vertex coloring on comparability graphs, and other combinatorial problems on comparability graphs and their complements.
The Graph Isomorphism Problem
, 1996
"... The graph isomorphism problem can be easily stated: check to see if two graphs that look differently are actually the same. The problem occupies a rare position in the world of complexity theory, it is clearly in NP but is not known to be in P and it is not known to be NPcomplete. Many subdiscipli ..."
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Cited by 64 (0 self)
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The graph isomorphism problem can be easily stated: check to see if two graphs that look differently are actually the same. The problem occupies a rare position in the world of complexity theory, it is clearly in NP but is not known to be in P and it is not known to be NPcomplete. Many subdisciplines of mathematics, such as topology theory and group theory, can be brought to bear on the problem, and yet only for special classes of graphs have polynomialtime algorithms been discovered. Incongruently, this problem seems very easy in practice. It is almost always trivial to check two random graphs for isomorphism, and fast hardware implementations exists for application domains such as image processing. This paper is mostly a survey of related work in the graph isomorphism field. We examine the problem from many angles, mirroring the multifaceted nature of the literature. We survey complexity results for the graph isomorphism problem, and discuss some of the classes of graphs which hav...
A Compact Data Structure and Parallel Algorithms for Permutation Graphs
 WG '95 21ST WORKSHOP ON GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE
, 1995
"... . Starting from a permutation of f0; : : : ; n \Gamma 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimens ..."
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Cited by 2 (2 self)
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. Starting from a permutation of f0; : : : ; n \Gamma 1g we compute in parallel with a workload of O(n log n) a compact data structure of size O(n log n). This data structure allows to obtain the associated permutation graph and the transitive closure and reduction of the associated order of dimension 2 efficiently. The parallel algorithms obtained have a workload of O(m + n log n) where m is the number of edges of the permutation graph. They run in time O(log 2 n) on a CREW PRAM. 1 Introduction Permutation graphs are combinatorial objects that found a lot of attention in recent years. This interest led to many results under a structural point of view as well as algorithmically, see [1, 2, 6, 7, 8]. By definition permutation graphs have a compact encoding of size n, where n is the number of vertices. In sequential computing model, it is possible to pass from the graph to the permutation and vice versa with a workload of O(n 2 ). For parallel computing this has been open up to no...
Towards a Promising Edge Classification Algorithm for the Graph Isomorphism Problem
, 2013
"... For over three decades the Graph Isomorphism (GI) problem has been extensively studied by many researchers in algorithms and complexity theory. To date, there is no formal proof to classify this problem to be in the class P or the class NP. In this paper, evidence had been proposed of the existing o ..."
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For over three decades the Graph Isomorphism (GI) problem has been extensively studied by many researchers in algorithms and complexity theory. To date, there is no formal proof to classify this problem to be in the class P or the class NP. In this paper, evidence had been proposed of the existing of polynomial time algorithm based on edge classification which can be used to prove that GI is rather in the class P.