Treewidth of planar graphs: connections with duality

International audienceRobertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result

On treewidth approximations

We introduce a natural heuristic for approximating the treewidth of graphs. We prove that this heuristic gives a constant factor approximation for the treewidth of graphs with bounded asteroidal number. Using a different technique, we give a $O(\log k)$ approximation algorithm for the treewidth of arbitrary graphs, where $k$ is the treewidth of the input graph

Ordres représentables par des translations de segments dans le plan

Certains ensembles ordonnes peuvent être représentés par des translations de figures convexes du plan. Il est prouvé, dans cette note, que les ordres sans N et les ordres d'intervalles admettent une telle représentation mais que le nombre de directions ne peut être borné. En revanche, les arbres sont représentables avec seulement deux directions. Pour toutes ces classes, les convexes utilisés sont des segments

Contiguity orders

This paper is devoted to the study of contiguity orders i.e. orders having a linear extension extension L such that all upper (or lower) cover sets are intervals of L. This new class is a strict generalization of both interval orders and N-free orders and is linearly recognizable. It is proved that computing the number of contiguity extensions is #P-complete and that the dimension of height one contiguity orders is polynomially tractable. Moreover the membership is a comparability invariant on bi-contiguity orders. Finally for strong-contiguity orders the calculation of the dimension is NP-complete

Listing all potential maximal cliques of a graph

A potential maximal clique of a graph is a vertex set that induces a maximal clique in some minimal triangulation of that graph. It is known that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We show here that the potential maximal cliques of a graph can be generated in polynomial time in the number of minimal separators of the graph. Thus, the treewidth and the minimum fill-in are polynomially tractable for all graphs with polynomial number of minimal separators.Une clique maximale potentielle d'un graphe est un ensemble de sommets qui induit une clique maximale dans au moins une triangulation minimale de ce graphe. Il a été prouvé que si ces objets peuvent être énumérés en temps polynomial pour une classe de graphes, la largeur arborescente et la complétion minimale sont calculables en temps polynomial pour ces graphes. Nous montrons ici que les cliques maximales potentielles d'un graphe peuvent être générées en temps polynomial par rapport au nombre de ses séparateurs minimaux. En conséquence, la largeur arborescente et la complétion minimale sont calculables en temps polynomial pour tous les graphes ayant un nombre polynomial de séparateurs minimaux

Treewidth and Minimum Fill-in: Grouping the Minimal Separators

We use the notion of potential maximal clique to characterize the maximal cliques appearing in minimal triangulations of a graph. We show that if these objects can be listed in polynomial time for a class of graphs, the treewidth and the minimum fill-in are polynomially tractable for these graphs. We prove that for all classes of graphs for which polynomial algorithms computing the treewidth and the minimum fill-in exist, we can list their potential maximal cliques in polynomial time. Our approach unies these algorithms. Finally we show how to compute in polynomial time the potential maximal cliques of weakly triangulated graphs, for which the treewidth and the minimum fill-in problems were open

Chordal embeddings of planar graphs

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved this conjecture in the affirmative, using algebraic techniques. We give a much shorter proof of this result

Chordal embeddings of planar graph

Robertson and Seymour conjectured that the treewidth of a planar graph and the treewidth of its geometric dual differ by at most one. Lapoire solved the conjecture in the affirmative, using algebraic techniques. We give here a much shorter proof of this result.Robertson et Seymour ont conjecturé que la largeur arborescente d'un graphe planaire et celle de son dual différent d'au plus un. Lapoire a prouvé cette conjecture en utilisant des outils algébriques. Nous donnons ici une preuve beaucoup plus courte de ce résulta