Duality Theorems for Blocks and Tangles in Graphs

Abstract

We prove a duality theorem applicable to a a wide range of specialisations, as well as to some generalisations, of tangles in graphs. It generalises the classical tangle duality theorem of Robertson and Seymour, which says that every graph either has a large-order tangle or a certain low-width tree-decomposition witnessing that it cannot have such a tangle.Our result also yields duality theorems for profiles and for k-blocks. This solves a problem studied, but not solved, by Diestel and Oum and answers an earlier question of Carmesin, Diestel, Hamann and Hundertmark. arXiv:1605.09139v3 [math.CO]

Tangle-Tree Duality: In Graphs, Matroids and Beyond

“…In [8], we show that Theorem 3.2 implies duality theorems for k-blocks and for any given subset of k-tangles.…”

“…An important example of S-trees are (irredundant) S-trees over stars: those over some F all of whose elements are stars of separations. 8 In such an S-tree (T, α) the map α preserves the natural partial ordering on E(T ) defined by letting (x, y) < (u, v) if {x, y} = {u, v} and the unique {x, y}-{u, v} path in T joins y to u (see Figure 2). 6 The tangles introduced by Robertson and Seymour [20] for graphs are, essentially, the T ktangles for the set T k of triples of oriented separations (A, B) of order less than some fixed k whose three 'small' sides A together cover the graph.…”

“…7 Trees have at least one node [6]. 8 For example, a tree-decomposition of width < w and adhesion < k of a graph is an S k -tree for the set S k of separations of order < k over the set Fw of stars 3 Tangle-tree duality in abstract separation systems…”

A unified treatment of linked and lean tree-decompositions

“…For example, more recently, Diestel, Eberenz and Erde [7] showed that there exist families of stars B k and P k ⊂ N − → S k , which are fixed under shifting, such that the existence of S k -trees over B k or P k is dual to the existence of a k-block or k-profile in the graph respectively (k-blocks and k-profiles are examples of what Diestel and Oum call "highly cohesive structures" which represent obstructions to low width, see [8]). They defined the profile-width and block-width of a graph G, which we denote by blw(G) and prw(G), to be the smallest k such that there is an S k -tree over B k or P k respectively.…”

On the Block Number of Graphs

“…A unified framework for duality theorems for width-parameters in graphs and matroids was developed by Diestel and Oum [12,13]. Based on this framework, Diestel, Eberenz and Erde [11] proved a duality theorem for k-blocks and described a class T k of tree-decompositions such that a graph has no k-block if and only if it has a tree-decomposition in T k . The only downside is that T k is given rather abstractly and thus seems difficult to work with.…”