Unifying Duality Theorems for Width Parameters in Graphs and Matroids (Extended Abstract)

Abstract

We prove a general duality theorem for tangle-like dense objects in combinatorial structures such as graphs and matroids. This paper continues, and assumes familiarity with, the theory developed in [6].

A unified treatment of linked and lean tree-decompositions

“…It is observed in [9] that many existing ways of decomposing graphs can be expressed in this framework if we take − → S to be the universe of separations of a graph. For example given a star σ = {(A 1 , B 1 ), (A 2 , B 2 ), .…”

“…then it is shown in [9] that a graph G has a tree-decomposition of width < k − 1 if and only if there exists an S k -tree over F k . More examples will be given later in Section 5 when we apply Theorems 3.2 and 4.1 to existing types of tree-decomposition.…”

“…It essentially this property of a family of stars F that is used in [9] to prove a duality theorem for S k -trees over F . It may seem like a strong property to hold, but in fact it is seen to hold in a large number of cases corresponding to known width-parameters of graphs such as branch-width, tree-width (see Lemma 2.14), and path-width.…”

“…It can be shown that a branch-decomposition of λ is equivalent to an S-tree over a set T of stars which is fixed under shifting (See the proof of [ [9], Lemma 4.3]), and that the width of this branch-decomposition is the smallest k such that the S-tree is an S k -tree. Furthermore, in this way the definition of linked given in [11] Proof.…”

“…Given a star σ ∈ N − → S let us write n(σ) for the cardinality of the multiset σ. Diestel and Oum [9] showed that for the families of stars…”