Results 1  10
of
12
A simple lineartime modular decomposition algorithm for graphs, using order extension
, 2004
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Graph Decompositions and Factorizing Permutations
 Discrete Mathematics and Theoretical Computer Science
, 1997
"... A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propo ..."
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Cited by 17 (11 self)
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A factorizing permutation of a given undirected graph is simply a permutation of the vertices in which all decomposition sets appear to be factors. Such a concept seems to play a central role in recent papers dealing with graph decomposition. It is applied here for modular decomposition and we propose a linear algorithm that computes the whole decomposition tree when a factorizing permutation is provided. This algorithm can be seen as a common generalization of Ma and Hsu [9, 8] for modular decomposition of chordal graphs and Habib, Huchard and Spinrad [7] for inheritance graphs decomposition. It also suggests many new decomposition algorithms for various notions of graph decompositions.
A survey on Algorithmic Aspects of Modular Decomposition
, 2009
"... The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a larger number of combinatorial optimi ..."
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Cited by 16 (2 self)
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The modular decomposition is a technique that applies but is not restricted to graphs. The notion of module naturally appears in the proofs of many graph theoretical theorems. Computing the modular decomposition tree is an important preprocessing step to solve a larger number of combinatorial optimization problems. Since the first polynomial time algorithm in the early 70’s, the algorithmic of the modular decomposition has known an important development. This paper survey the ideas and techniques that arose from this line of research.
Faster Query Answering in Probabilistic Databases using ReadOnce Functions
"... A boolean expression is in readonce form if each of its variables appears exactly once. When the variables denote independent events in a probability space, the probability of the event denoted by the whole expression in readonce form can be computed in polynomial time (whereas the general problem ..."
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Cited by 7 (2 self)
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A boolean expression is in readonce form if each of its variables appears exactly once. When the variables denote independent events in a probability space, the probability of the event denoted by the whole expression in readonce form can be computed in polynomial time (whereas the general problem for arbitrary expressions is #Pcomplete). Known approaches to checking readonce property seem to require putting these expressions in disjunctive normal form. In this paper, we tell a better story for a large subclass of boolean event expressions: those that are generated by conjunctive queries without selfjoins and on tupleindependent probabilistic databases. We first show that given a tupleindependent representation and the provenance graph of an SPJ query plan without selfjoins, we can, without using the DNF of a result event expression, efficiently compute its cooccurrence graph. From this, the readonce form can already, if it exists, be computed efficiently using existing techniques. Our second and key contribution is a complete, efficient, and simple to implement algorithm for computing the readonce forms (whenever they exist) directly, using a new concept, that of cotable graph, which can be significantly smaller than the cooccurrence graph.
Probe Ptolemaic Graphs
, 2008
"... Given a class of graphs, G, a graph G is a probe graph of G if its vertices can be partitioned into two sets, P (the probes) and N (the nonprobes), where N is an independent set, such that G can be embedded into a graph of G by adding edges between certain nonprobes. In this paper we study the probe ..."
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Cited by 1 (0 self)
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Given a class of graphs, G, a graph G is a probe graph of G if its vertices can be partitioned into two sets, P (the probes) and N (the nonprobes), where N is an independent set, such that G can be embedded into a graph of G by adding edges between certain nonprobes. In this paper we study the probe graphs of ptolemaic graphs when the partition of vertices is unknown. We present some characterizations of probe ptolemaic graphs and show that there exists a polynomialtime recognition algorithm for probe ptolemaic graphs.
OPTIMIZATION PROBLEMS ON THRESHOLD GRAPHS
"... Abstract. During the last three decades, different types of decompositions have been processed in the field of graph theory. Among these we mention: decompositions based on the additivity of some characteristics of the graph, decompositions where the adjacency law between the subsets of the partitio ..."
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Abstract. During the last three decades, different types of decompositions have been processed in the field of graph theory. Among these we mention: decompositions based on the additivity of some characteristics of the graph, decompositions where the adjacency law between the subsets of the partition is known, decompositions where the subgraph induced by every subset of the partition must have predeterminate properties, as well as combinations of such decompositions. In this paper we characterize threshold graphs using the weakly decomposition, determine: density and stability number, Wiener index and Wiener polynomial for threshold graphs.
A Simple LinearTime Recognition Algorithm for Weakly QuasiThreshold Graphs
, 2011
"... Weakly quasithreshold graphs form a proper subclass of the wellknown class of cographs by restricting the join operation. In this paper we characterize weakly quasithreshold graphs by a finite set of forbidden subgraphs: the class of weakly quasithreshold graphs coincides with the class of {P4, ..."
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Weakly quasithreshold graphs form a proper subclass of the wellknown class of cographs by restricting the join operation. In this paper we characterize weakly quasithreshold graphs by a finite set of forbidden subgraphs: the class of weakly quasithreshold graphs coincides with the class of {P4, co(2P3)}free graphs. Moreover we give the first lineartime algorithm to decide whether a given graph belongs to the class of weakly quasithreshold graphs, improving the previously known running time. Based on the simplicity of our recognition algorithm, we can provide certificates of membership (a structure that characterizes weakly quasithreshold graphs) or nonmembership (forbidden induced subgraphs) in additional O(n) time. Furthermore we give a lineartime algorithm for finding the largest induced weakly quasithreshold subgraph in a cograph.
Hadwiger Number of Graphs with Small Chordality
, 2014
"... The Hadwiger number of a graph G is the largest integer h such that G has the complete graph Kh as a minor. We show that the problem of determining the Hadwiger number of a graph is NPhard on cobipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs ..."
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The Hadwiger number of a graph G is the largest integer h such that G has the complete graph Kh as a minor. We show that the problem of determining the Hadwiger number of a graph is NPhard on cobipartite graphs, but can be solved in polynomial time on cographs and on bipartite permutation graphs. We also consider a natural generalization of this problem that asks for the largest integer h such that G has a minor with h vertices and diameter at most s. We show that this problem can be solved in polynomial time on ATfree graphs when s ≥ 2, but is NPhard on chordal graphs for every fixed s ≥ 2.