{
  "id": "enwiki-001354296200",
  "slug": "graph-structure-theorem",
  "title": "Graph structure theorem",
  "qid": "",
  "lead": "In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its proof is very long and involved. and are surveys accessible to nonspecialists, describing the theorem and its consequences.",
  "body": "In mathematics, the graph structure theorem is a major result in the area of graph theory. The result establishes a deep and fundamental connection between the theory of graph minors and topological embeddings. The theorem is stated in the seventeenth of a series of 23 papers by Neil Robertson and Paul Seymour. Its proof is very long and involved. and are surveys accessible to nonspecialists, describing the theorem and its consequences. ==Refinements== Strengthened versions of the graph structure theorem are possible depending on the set of forbidden minors. For instance, when one of the graphs in is planar, then every -minor-free graph has a tree decomposition of bounded width; equivalently, it can be represented as a clique-sum of graphs of constant size. When one of the graphs in can be drawn in the plane with only a single crossing, then the -minor-free graphs admit a decomposition as a clique-sum of graphs of constant size and graphs of bounded genus, without vortices. A different strengthening is also known when one of the graphs in is an apex graph.",
  "categories": [
    "23118434",
    "23603337",
    "29264962",
    "29265318"
  ],
  "license": "CC BY-SA 4.0",
  "source_uri": "https://en.wikipedia.org/w/index.php?oldid=1354296200",
  "content_hash": "",
  "created_at": null,
  "updated_at": 1778848667.0,
  "claims": [],
  "contradictions": [],
  "confidence": 0.6,
  "csl": {
    "id": "enwiki-001354296200",
    "type": "webpage",
    "title": "Graph structure theorem",
    "author": [
      {
        "family": "Anticloud FZ LLE",
        "literal": "Anticloud FZ LLE"
      }
    ],
    "container-title": "encyclopegia.org",
    "publisher": "Anticloud FZ LLE",
    "publisher-place": "Dubai, United Arab Emirates",
    "URL": "https://encyclopegia.org/a/graph-structure-theorem",
    "issued": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "accessed": {
      "date-parts": [
        [
          2026,
          11,
          15
        ]
      ]
    },
    "license": "CC BY-SA 4.0",
    "DOI": "eo1:enwiki-001354296200",
    "isBasedOn": "https://en.wikipedia.org/w/index.php?oldid=1354296200",
    "note": "Immutable snapshot enwiki-001354296200"
  }
}