Partitions versus sets : a case of duality

Laurent Lyaudet (1) , Frédéric Mazoit (2) , Stéphan Thomassé (3)

1 LIP - Laboratoire de l'Informatique du Parallélisme
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique
3 ALGCO - Algorithmes, Graphes et Combinatoire

In a recent paper, Amini et al. introduce a general framework to prove duality theorems between special decompositions and their dual combinatorial object. They thus unify all known ad-hoc proofs in one single theorem. While this unification process is definitely good, their main theorem remains quite technical and does not give a real insight of why some decompositions admit dual objects and why others do not. The goal of this paper is both to generalise a little this framework and to give an enlightening simple proof of its central theorem.

Keywords: tree-width; bramble; duality; submodularity

Domains: Discrete Mathematics [cs.DM]

Laurent Lyaudet, Frédéric Mazoit, Stéphan Thomassé. Partitions versus sets : a case of duality. European Journal of Combinatorics, 2009, pp.1-7. ⟨10.1016/j.ejc.2009.09.004⟩. ⟨hal-00345894v3⟩