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{
    "pk": 64941,
    "title": "Refining trees of tangles in abstract separation systems: inessential parts",
    "subtitle": null,
    "abstract": "Robertson and Seymour proved two fundamental theorems about tangles in graphs: the tree-of-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 \\(k\\)-tangle 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.\nErde 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.\nThe 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.\n \nMathematics Subject Classifications: 05C83, 05C40, 06A07\n \nKeywords: Tree of tangles, tangle-tree duality, abstract separation system, submodularity, canonical",
    "language": "en",
    "license": {
        "name": "Creative Commons Attribution 4.0",
        "short_name": "CC BY 4.0",
        "text": "Attribution — You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.\n\nNo additional restrictions — You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits.",
        "url": "https://creativecommons.org/licenses/by/4.0"
    },
    "keywords": [
        {
            "word": "Tree of tangles"
        },
        {
            "word": "tangle-tree duality"
        },
        {
            "word": "abstract separation system"
        },
        {
            "word": "submodularity"
        },
        {
            "word": "canonical"
        }
    ],
    "section": "Research Articles",
    "is_remote": true,
    "remote_url": "https://escholarship.org/uc/item/0pt1052t",
    "frozenauthors": [
        {
            "first_name": "Sandra",
            "middle_name": "",
            "last_name": "Albrechtsen",
            "name_suffix": "",
            "institution": "Universität Hamburg, Department of Mathematics, Bundesstraße 55 (Geomatikum), 20146 Hamburg, Germany",
            "department": ""
        }
    ],
    "date_submitted": "2024-07-01T16:37:09+02:00",
    "date_accepted": "2024-07-01T16:37:09+02:00",
    "date_published": "2024-06-30T09:00:00+02:00",
    "render_galley": null,
    "galleys": [
        {
            "label": "",
            "type": "pdf",
            "path": "https://journalpub.escholarship.org/combinatorial_theory/article/64941/galley/49751/download/"
        }
    ]
}