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  "body": "\n## Statement\n\nThe finite graphs are well-quasi-ordered by the [[minor|VYQ32ZBPEE]] relation.\nThat is, for every infinite sequence $G_1, G_2, G_3, \\dots$ of finite graphs, there are indices $i \u003c j$ such that $G_i$ is a minor of $G_j$.\n\n## Remarks\n\nAn equivalent form: let $\\mathcal{C}$ be a class of finite graphs closed under isomorphism and under taking minors.\nThen there is a finite set $\\mathcal{F}$ of graphs such that a graph lies in $\\mathcal{C}$ if and only if no member of $\\mathcal{F}$ is a minor of it.\nFor planar graphs $\\mathcal{F} = \\set{K_5, K_{3,3}}$, by [[Wagner's theorem|1B2ZJPVV03]], and for forests $\\mathcal{F} = \\set{K_3}$.\nNeil Robertson and Paul Seymour proved the theorem in a long series of papers called Graph Minors, published between 1983 and 2004.\nThe proof rests on a structure theorem for graphs that exclude a fixed minor, built on tree decompositions and embeddings in surfaces.\nThe same project gives a polynomial-time algorithm that tests whether a fixed graph $H$ is a minor of an input graph.\nSo every minor-closed class can be recognized in polynomial time.\nThis proof of polynomial-time recognition is not constructive, since the theorem does not say how to find the set $\\mathcal{F}$.\n\n## Explanations\n\n### L1 human\n\nA minor of a graph is what remains after deleting and shrinking parts of it.\nThe theorem says that you cannot have an infinite list of graphs in which none is a minor of a later one.\nAs a result, every property that survives deleting and shrinking is described by a finite list of forbidden patterns.\n\n### L2 human\n\nFor trees, the analogous statement is [[Kruskal's tree theorem|XVHKCX4F5Y]].\nThe proof of the general case extends ideas from it through [[treewidth|CD2XB4Y1SF]].\nGraphs of bounded treewidth are handled like trees, and graphs of large treewidth contain large grid minors.\nThe theorem is one of the deepest results of combinatorics and gives finite obstruction sets for embeddability in any fixed surface.\n",
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