of [*only if*]{}: this is proved by induction on the number of vertices. Let Ulam’s conjecture be true. Since $N_{i1} = e(\Lambda_i)$ for all $i$, every non-empty three vertex graph is $\mathcal{N}$-matrix reconstructible. Now, let all graphs on at most $n$ vertices, where $n\geq 3$, be $\mathcal{N}$-matrix reconstructible. Let $G$ be a graph on $n+1$ vertices. By Lemma \[lem-kelly3\], the collection $\{\mathcal{N}(G-u); u\in VG, e(G-u) > 0\}$ is unambiguously determined by $\mathcal{N}(G)$. The number of empty graphs in $\mathcal{VD}(G)$ is 0 or 1, and is determined by Lemma \[lem-empty\]. Therefore, by induction hypothesis, $\mathcal{VD}(G)$ is uniquely determined. Now the result follows from the assumption that Ulam’s conjecture is true.

Since $\mathcal{N}(G)$ and $\mathcal{ELP}(G)$ are equivalent by Lemma \[lem-elp\_eq\_n\], we rephrase Proposition \[prop:equiv-un\] as follows.

\[prop:equiv-elp\] Ulam’s conjecture is true if and only if all graphs on three or more vertices are reconstructible from their abstract edge labelled posets.

We would like to point out that reconstructing $G$ from $\mathcal{N}(G)$ or from $\mathcal{ELP}(G)$ is not proved to be equivalent to reconstructing $G$ from $\mathcal{VD}(G)$. This poses a difficulty. For example, proving $\mathcal{N}$-matrix reconstructibility of disconnected graphs is as hard as Ulam’s conjecture, although disconnected graphs are known to be vertex reconstructible. 