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For $n\geq 5$, let $\mathcal {P}_n$ be the set of all isomorphism classes of graphs with n vertices. Give this set the poset structure given by $G \le H$ if and only if $G$ is a subgraph of $H$.

Is it true that $\mathcal {P}_n$ has no nontrivial automorphisms?

Remarks:

This follows if one can recognize a graph from the set of isomorphism classes of its edge-deleted subgraphs. However, since recognizing a graph from the set of isomorphism classes of its edge-deleted subgraphs is stronger than edge reconstruction, I'm wondering if there is an alternative way of proving this.

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    $\begingroup$ Maybe it is useful to add that $\cal P_4$ has a nontrivial automorphism. Did you ckeck for higher values, e.g. using the tables from maths.uq.edu.au/~pa/research/posets4to8.html? $\endgroup$
    – domotorp
    Commented Dec 31, 2013 at 5:46
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    $\begingroup$ $\mathcal P_5$ can pretty easily be checked by hand. As for anything larger, I'm not sure. $\endgroup$
    – Wade Hann-Caruthers
    Commented Dec 31, 2013 at 6:30
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    $\begingroup$ Of course there is in fact no need to check, as it anyhow follows from the set edge reconstruction conjecture but I could not find anything about for what values that has been checked. $\endgroup$
    – domotorp
    Commented Dec 31, 2013 at 11:48
  • $\begingroup$ Probably has something to do with: math.stackexchange.com/questions/3052384/… $\endgroup$
    – Daniel Donnelly
    Commented May 21 at 2:18

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