Wikipedia's article on the Reconstruction Conjecture mentions that the conjecture is false for digraphs, and refers to two papers by Stockmeyer. As far as I can see, none of the counter-examples in those papers are acyclic, so my question is

Can a directed acyclic graph be reconstructed from its deck of vertex-deleted subgraphs?

One has to assume the graph has at least $5$ vertices to avoid certain small cases. (Edit: For $4$ vertices, see Julian's example below.) Acyclic tournaments are reconstructible according to the references.

The question has an equivalent reformulation in representation theory:

Let $Q$ be a directed acyclic graph as above, and let $k$ be an algebraically closed field. Can the path algebra $\Lambda=kQ$ be reconstructed from its deck of vertex-deleted quotients $\Lambda/\Lambda e \Lambda$?

deckis just a multiset. If theDeckReconstruction Conjecture is equivalent to theSetReconstruction Conjecture I think still is an open question. $\endgroup$ – Dag Oskar Madsen Dec 19 '14 at 10:22