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18
Linear time solvable optimization problems on graphs of bounded cliquewidth, Extended abstract
 Graph Theoretic Concepts in Computer Science, 24th International Workshop, WG ’98, Lecture Notes in Computer Science
, 1998
"... Abstract. Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of treewidth at most k, i.e., that have tree decompositions of width at most k, where k is fixed, ..."
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Cited by 113 (20 self)
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Abstract. Hierarchical decompositions of graphs are interesting for algorithmic purposes. There are several types of hierarchical decompositions. Tree decompositions are the best known ones. On graphs of treewidth at most k, i.e., that have tree decompositions of width at most k, where k is fixed, every decision or optimization problem expressible in monadic secondorder logic has a linear algorithm. We prove that this is also the case for graphs of cliquewidth at most k, where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications. We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with “too many ” induced paths with four vertices. 1.
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 Closure of Monadic NP
 Journal of Computer and System Sciences
, 1997
"... It is a wellknown result of Fagin that the complexity class NP coincides with the class of ..."
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Cited by 21 (0 self)
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It is a wellknown result of Fagin that the complexity class NP coincides with the class of
Fast and simple algorithms for recognizing chordal comparability graphs and interval graphs
 SIAM Journal on Computing
, 1999
"... Abstract. In this paper, we present a lineartime algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a lineartime algorithm for transitive orientation on chordal comparability graphs, which reduces the complexity of chordal comparability recognition from O( ..."
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Cited by 16 (3 self)
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Abstract. In this paper, we present a lineartime algorithm for substitution decomposition on chordal graphs. Based on this result, we develop a lineartime algorithm for transitive orientation on chordal comparability graphs, which reduces the complexity of chordal comparability recognition from O(n 2)toO(n+m). We also devise a simple lineartime algorithm for interval graph recognition where no complicated data structure is involved. Key words. chordal graph, triangulated graph, interval graph, analysis of algorithms, graph theory, substitution decomposition, modular decomposition, cyclefree poset, transitive orientation, graph partitioning, cardinality lexicographic ordering, graph recognition
The Homogeneous Set Sandwich Problem
, 1998
"... The graph sandwich problem for property \Phi is defined as follows: Given two graphs G 1 = (V; E 1 ) and G 2 = (V; E 2 ) such that E 1 ` E 2 , is there a graph G = (V; E) such that E 1 ` E ` E 2 which satisfies property \Phi? We present a polynomialtime algorithm for solving the graph sandwich pro ..."
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Cited by 7 (4 self)
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The graph sandwich problem for property \Phi is defined as follows: Given two graphs G 1 = (V; E 1 ) and G 2 = (V; E 2 ) such that E 1 ` E 2 , is there a graph G = (V; E) such that E 1 ` E ` E 2 which satisfies property \Phi? We present a polynomialtime algorithm for solving the graph sandwich problem, when property \Phi is "to contain a homogeneous set". The algorithm presented also provides the graph G and a homogeneous set in G in case it exists.
On the CliqueWidth of Graphs with Few P 4 s
, 1998
"... Babel and Olariu (1995) introduced the class of (q; t) graphs in which every set of q vertices has at most t distinct induced P 4 s. Graphs of cliquewidth at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k expressions based on graph operati ..."
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Cited by 3 (1 self)
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Babel and Olariu (1995) introduced the class of (q; t) graphs in which every set of q vertices has at most t distinct induced P 4 s. Graphs of cliquewidth at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k expressions based on graph operations which use k vertex labels. In this paper we study the cliquewidth of the (q; t) graphs, for almost all possible combinations of q and t. On one hand we show that every (q; q \Gamma 3) graph for q 7, has clique width q and a qexpression defining it can be obtained in linear time. On the other hand we show that this result does not hold for the class of (q; q) graphs for any q, and for the class of (q; q \Gamma 3) graphs for q 6. More precisely, we show that for every q, for every n 2 N there is a graph H n which is a (q; q) graph having n vertices and the cliquewidth of H n is at least ( p n=3q)=3q. Partially supported by a Grant of the Israeli Ministry of Science for Fr...
Enumeration of PinPermutations
, 2008
"... In this paper, we study the class of pinpermutations, that is to say of permutations having a pin representation. This class has been recently introduced in [16], where it is used to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, de ..."
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In this paper, we study the class of pinpermutations, that is to say of permutations having a pin representation. This class has been recently introduced in [16], where it is used to find properties (algebraicity of the generating function, decidability of membership) of classes of permutations, depending on the simple permutations this class contains. We give a recursive characterization of the substitution decomposition trees of pinpermutations, which allows us to compute the generating function of this class, and consequently to prove, as it is conjectured in [18], the rationality of this generating function. Moreover, we show that the basis of the pinpermutation class is infinite.
Polynomial Time Recognition of P 4 structure
"... A P4 is a set of four vertices of a graph that induces a chordless path; the P4structure of a graph is the set of all P4 's. Vasek Chvatal asked if there is a polynomial time algorithm to determine whether an arbitrary fouruniform hypergraph is the P4structure of some graph. The answer is yes; we ..."
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Cited by 1 (0 self)
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A P4 is a set of four vertices of a graph that induces a chordless path; the P4structure of a graph is the set of all P4 's. Vasek Chvatal asked if there is a polynomial time algorithm to determine whether an arbitrary fouruniform hypergraph is the P4structure of some graph. The answer is yes; we present such an algorithm.
Cograph Recognition Algorithm Revisited and Online Induced P 4 Search
, 1994
"... . In 1985, Corneil, Perl and Stewart [CPS85] gave a linear incremental algorithm to recognize cographs (graphs with no induced P4 ). When this algorithm stops, either the initial graph is a cograph and the cotree of the whole graph has been built, or the initial graph is not a cograph and this algo ..."
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. In 1985, Corneil, Perl and Stewart [CPS85] gave a linear incremental algorithm to recognize cographs (graphs with no induced P4 ). When this algorithm stops, either the initial graph is a cograph and the cotree of the whole graph has been built, or the initial graph is not a cograph and this algorithm ends up with a vertex v and a cotree cot such that v cannot be inserted in cot; so the input graph must contain a P4 . In many applications such as graph decomposition [Cou93, CH93a, CH93b, CH94, EGMS94, Spi92, MS94], transitive orientation [Spi83, ST94], not only the existence but a P4 is also explicitly needed. In this paper, we present a new characterization of cograph in terms of its modular structure. This characterization yields a structural labeling of the cotree for incremental cograph recognition, and we show how to go from this labeling to the Corneil et al. one's. Furthermore, we show how to adapt this algorithm in order to produce a P4 in case of failure when adding a new v...