Structural submodularity and tangles in abstract separation systems

Diestel, Reinhard; Erde, Joshua; Weißauer, Daniel

doi:10.48550/arxiv.1805.01439

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“…In this section we give a short overview of the important objects, definitions and theorems that we are going to use later. For a more detailed introduction to abstract separation systems and their tangles we refer the reader to [8] and [6]. The definitions are taken from [5,6,8,9,11].…”

Section: Abstract Separation Systems and Tanglesmentioning

confidence: 99%

“…For a more detailed introduction to abstract separation systems and their tangles we refer the reader to [8] and [6]. The definitions are taken from [5,6,8,9,11]. We use the basic graph-theoretic notions from [4].…”

Section: Abstract Separation Systems and Tanglesmentioning

confidence: 99%

“…Submodularity makes it possible to generalize the fundamental theorems of the tangle theory of graphs, such as the tree-of-tangles theorem and the tangle-tree duality theorem, to abstract separation systems [8,Theorem 6 and 4]. In this more general setting, tangles are distinguished by a tree of tangles which generalizes the concept of tree-decompositions.…”

Section: Introductionmentioning

confidence: 99%

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Refining trees of tangles in abstract separation systems I: Inessential parts

Sandra¹

2023

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Robertson and Seymour proved two fundamental theorems about tangles in graphs: the treeof-tangles theorem, which says that every graph has a tree-decomposition such that distinguishable tangles live in different nodes of the tree, and the tangle-tree duality theorem, which says that graphs without a ktangle have a tree-decomposition that witnesses the non-existence of such tangles, in that k-tangles would have to live in a node but no node is large enough to accommodate one.Erde combined these two fundamental theorems into one, by constructing a single tree-decomposition such that every node either accommodates a single k-tangle or is too small to accommodate one. Such a tree-decomposition thus shows at a glance how many k-tangles a graph has and where they are.The two fundamental theorems have since been extended to abstract separation systems, which support tangles in more general discrete structures. In this paper we extend Erde's unified theorem to such general systems.

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