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  "body": "\n## Statement\n\nFor a finite [[graph|0JD5C82QMB]] $G$ and a vertex $v$ of $G$, let $G - v$ be the [[induced subgraph|0V05FC4GKQ]] on the vertices other than $v$.\nThe deck of $G$ is the multiset of the [[isomorphism|00CT0PTH74]] classes of the graphs $G - v$, one for each vertex $v$.\nConjecture: if $G$ and $H$ are finite graphs with at least $3$ vertices and the same deck, then $G$ and $H$ are isomorphic.\n\n## Remarks\n\nThe bound on the number of vertices is needed.\nThe graph with $2$ vertices and one edge and the graph with $2$ vertices and no edge have the same deck, made of two single vertices.\nThe conjecture is due to Kelly and Ulam, and it is open.\nMany invariants can be read off the deck, for example the number of edges and the degree sequence.\nSeveral classes of graphs are known to be reconstructible from their decks, among them regular graphs, disconnected graphs and trees.\nComputer searches have confirmed the conjecture for all graphs with few vertices.\n\n## Explanations\n\n### L1 human\n\nTake a graph and make one copy of it for each vertex, with that vertex and its edges erased.\nShuffle the copies and forget which vertex was erased in each.\nThe conjecture says that, as soon as there are at least three vertices, the shuffled pile of copies always determines the original graph.\n",
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      "first_stated": "1942",
      "posed_by": "Paul J. Kelly and Stanisław Ulam"
    },
    "kind": "conjecture",
    "msc": [
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    "name": "Graph reconstruction conjecture",
    "same_as": {
      "wikidata": "Q7302789",
      "wikipedia": "Reconstruction conjecture"
    },
    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Reconstruction_conjecture"
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    "tag": "XRBPS2QYE3",
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  "kind": "conjecture",
  "link": "https://math2.tamnd.com/node/XRBPS2QYE3/",
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  "markdown": "https://math2.tamnd.com/node/XRBPS2QYE3.md",
  "msc": [
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  "name": "Graph reconstruction conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/XRB/XRBPS2QYE3.md",
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  "url": "https://math2.tamnd.com/wiki/graph-reconstruction-conjecture/",
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    "tag": "XRBPS2QYE3",
    "kind": "conjecture",
    "level": "W0",
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