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25
Linear time solvable optimization problems on graphs of bounded cliquewidth
, 2000
"... 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 dec ..."
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Cited by 170 (22 self)
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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.
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 111 (12 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 29 (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 19 (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 11 (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.
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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Cited by 7 (5 self)
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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.
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...
Polynomial Time Recognition of P4structure
"... 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 ye ..."
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Cited by 2 (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.
A fully dynamic algorithm for the recognition of P4sparse graphs
 PROC. 32ND WORKSHOP ON GRAPHTHEORETIC CONCEPTS IN COMPUTER SCIENCE (WG 2006), LNCS 4271
, 2006
"... In this paper, we solve the dynamic recognition problem for the class of P4sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a P4sparse graph, and if yes, to update a representation of the graph. Our approach relies on main ..."
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Cited by 2 (0 self)
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In this paper, we solve the dynamic recognition problem for the class of P4sparse graphs: the objective is to handle edge/vertex additions and deletions, to recognize if each such modification yields a P4sparse graph, and if yes, to update a representation of the graph. Our approach relies on maintaining the modular decomposition tree of the graph, which we use for solving the recognition problem. We establish properties for each modification to yield a P4sparse graph and obtain a fully dynamic recognition algorithm which handles edge modifications in O(1) time and vertex modifications in O(d) time for a vertex of degree d. Thus, our algorithm implies an optimal edgesonly dynamic algorithm and a new optimal incremental algorithm for P4sparse graphs.