{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $G$ be a finite [[graph|0JD5C82QMB]].\nThen $G$ is [[planar|029JQEK15B]] if and only if neither the [[complete graph|03WKB2H474]] $K_5$ nor the [[complete bipartite graph|0V0GC7BQAN]] $K_{3,3}$ is a [[minor|VYQ32ZBPEE]] of $G$.\n\n## Remarks\n\nDeleting vertices or edges and contracting edges keep a plane drawing plane, so every minor of a planar graph is planar.\nSince $K_5$ and $K_{3,3}$ are not planar, a planar graph has neither as a minor.\nFor the converse, if $G$ is not planar, [[Kuratowski's theorem|0J6G5NJC5R]] gives a subgraph that is a subdivision of $K_5$ or of $K_{3,3}$.\nContracting the subdivided paths of this subgraph back to single edges gives $K_5$ or $K_{3,3}$ as a minor.\nFor example, the [[Petersen graph|0NR95CTFCZ]] is not planar: contracting its five spokes gives $K_5$.\nKlaus Wagner published the theorem in 1937.\n\n## Explanations\n\n### L1 human\n\nA graph is planar when you can draw it without crossings.\nWagner's theorem says that the only obstacles are the five-vertex complete graph and the three-houses-three-utilities graph, found inside $G$ after deleting and shrinking parts.\nSo to prove a graph is not planar, it is enough to exhibit one of these two inside it.\n\n### L2 human\n\nWagner's theorem is the minor version of Kuratowski's theorem.\nIt shows that the class of planar graphs is closed under minors and is described by exactly two forbidden minors.\nThe [[Robertson–Seymour theorem|WP3GDNWJJW]] says that every minor-closed class of graphs has such a finite description.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "029JQEK15B",
      "VYQ32ZBPEE"
    ],
    "kind": "statement",
    "msc": [
      "05C10",
      "05C83"
    ],
    "name": "Wagner's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7959587",
      "wikipedia": "Wagner's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Wagner's_theorem"
      }
    ],
    "tag": "1B2ZJPVV03",
    "uses": [
      "0JD5C82QMB",
      "029JQEK15B",
      "VYQ32ZBPEE",
      "03WKB2H474",
      "0V0GC7BQAN"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/1B2ZJPVV03/",
  "links": [
    {
      "field": "about",
      "target": "029JQEK15B"
    },
    {
      "field": "about",
      "target": "VYQ32ZBPEE"
    },
    {
      "field": "uses",
      "target": "0JD5C82QMB"
    },
    {
      "field": "uses",
      "target": "029JQEK15B"
    },
    {
      "field": "uses",
      "target": "VYQ32ZBPEE"
    },
    {
      "field": "uses",
      "target": "03WKB2H474"
    },
    {
      "field": "uses",
      "target": "0V0GC7BQAN"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/1B2ZJPVV03.md",
  "msc": [
    "05C10",
    "05C83"
  ],
  "name": "Wagner's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/1B2/1B2ZJPVV03.md",
  "status": null,
  "tag": "1B2ZJPVV03",
  "uri": "m2://1B2ZJPVV03",
  "url": "https://math2.tamnd.com/wiki/wagners-theorem/",
  "writing": {
    "tag": "1B2ZJPVV03",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}