Constructing brambles

Mathieu Chapelle (1) , Frédéric Mazoit (2) , Ioan Todinca (1)

1 LIFO - Laboratoire d'Informatique Fondamentale d'Orléans
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

Given an arbitrary graph G and a number k, it is well-known by a result of Seymour and Thomas that G has treewidth strictly larger than k if and only if it has a bramble of order k + 2. Brambles are used in combinatorics as certificates proving that the treewidth of a graph is large. From an algorithmic point of view there are several algorithms computing tree-decompositions of G of width at most k, if such decompositions exist and the running time is polynomial for constant k. Nevertheless, when the treewidth of the input graph is larger than k, to our knowledge there is no algorithm constructing a bramble of order k + 2. We give here such an algorithm, running in O(n^{k+4} ) time. Moreover, for classes of graphs with polynomial number of minimal separators, we define a notion of compact brambles and show how to compute compact brambles of order k + 2 in polynomial time, not depending on k.

Domains: Discrete Mathematics [cs.DM]

Mathieu Chapelle, Frédéric Mazoit, Ioan Todinca. Constructing brambles. International Symposium on Mathematical Foundations of Computer Science (MFCS), 2009, France. pp.223-234, ⟨10.1007/978-3-642-03816-7⟩. ⟨hal-00407682⟩