Vincent Bouchitté's research while affiliated with Ecole normale supérieure de Lyon and other places

... Finally, S is a minimal separator if S is a u,v-minimal separator for some pair of vertices u and v. Minimal separators have a tremendous role in the design of graph algorithms, both directly, such as in the structural characterization of chordal graphs [5] but also indirectly in optimization algorithms for graph separation and routing problems (for example [17,16,21]). The theory of potential maximal cliques, developed by Bouchitté and Todinca [4] implies that a several fundamental graph problems, such as computing the treewidth and minimum fill in of a graph G can be done in time polynomial in the number of vertices of G and the number of minimal separators in G. Lokshtanov [15] showed that the same result holds for computing the tree-length of the graph G, while Fomin et al. [10] proved a general result that showed that a whole class of problems (including e.g. maximum independent set and minimum feedback vertex set) can be solved in time polynomial in the number of vertices and minimal separators of the graph. ...

... Complexité Dans [BH89] les auteurs conjecturent que le problème de comptage des extensions linéaires d'un ordre sans cycle est de complexité polynomiale. Nous avons montré que ce calcul pouvait se faire en évaluant une formule intégrale de taille linéaire en la taille de l'ordre en entrée. ...

... 18 V. Bouchitt e, A. Hilali, R. J egou, J.X. Rampon With Proposition 2 and condition (iv) of Theorem 10, we get a new proof of the lower{contiguity of interval orders stated in 4] . Moreover, adding condition (iii), we immediately have: P = (X; P ) is a lattice ii any two element subset of X has an innmum and a supremum in P. The innmum (resp. ...

... We recall that if y x, then there exists a greedy linear extension that puts x before y [2]. An α-greedy balanced pair in P = (V, ≤) is a pair (x, y) of elements of V such that the ratio of greedy linear extensions of P that put x before y among all greedy linear extensions, denoted GP P (x < y), is in the real interval [α, 1 − α]. ...

... l Using the proof of the claim: "For any positive integer m, there is an ordered set with no m-directional representation" given in [S], we show in [2] that for every integer k there exists an interval order which is not k-directional. l These new notions which can model separability problems in computational geometry seem to be an attractive domain for further investigations. ...

... , h k ; h 1 , . . . , h k } whose only edges are h i h j when there is an arc from h i to h j in H ; also see [1]. If H is transitively oriented, then [4] shows that H is weakly chordal if and only if its bipartite transform is chordal bipartite. ...

... Maximizing the jump number has many efficient, and some elegant, polynomial time algorithms (this is the bump number problem) [10]. Minimizing the number of jumps is called the jump number problem, and is NP-hard [11,1]. Exact algorithms for jump number have recently been found to run in O(1.824 n ) [5], slightly improving the previous bound of Kratsch and Kratsch's algorithm running in O(1.8638 n ) [6]; the latter also provide a O(1.7593 n )-time algorithm for interval posets. ...

... First, by leveraging on classic results from [39], we show that the problem of constructing a ρ-consistent path decomposition of approximately minimum width for the cocomparability graph G ρ of a given partial-order ρ is fixed-parameter tractable with respect to the pathwidth of G ρ . While it was known that the pathwidth and the ρ-consistent pathwidth of G ρ are always the same [3], and that there were fixed-parameter tractable algorithms for computing path decompositions of approximately minimum width due to structural properties of cocomparability graphs [8], the problem of computing such a decomposition satisfying the additional ρ-consistent requirement was open [3]. ...

... No hardness of approximation is known and not even the possibility of a polynomial-time approximation scheme for Treewidth has been ruled out. In many important special classes of graphs, such as planar graphs (Seymour & Thomas, 1994), asteroidal triple-free graphs (Bouchitté & Todinca, 2003), and H-minor-free graphs (Feige et al., 2005), constant factor approximations are known, but the general case has remained elusive. ...

... The second phase of the PMC framework has already been formulated for all of the problems that we consider [15,17,18,21,35], so our O * (3 cc ) time algorithms follow from an O * (3 cc ) time PMC enumeration algorithm that we give. This algorithm is based on the Bouchitté-Todinca algorithm [6]. We achieve the O * (3 cc ) bound by novel characterizations of minimal separators and PMCs with respect to an edge clique cover. ...