disconnected graphs that are not $\mathcal{N}$-matrix reconstructible.

The following result is proved along the lines of Lemma \[lem-2g2h\].

Ulam’s conjecture is true if and only if the edge labelled poset of each graph has only the trivial automorphism.

The proof of [*only if*]{} is done by contradiction. Suppose that $\mathcal{ELP}(G)$ has a nontrivial automorphism $\sigma$. Then there are nonisomorphic induced subgraphs $\Lambda_i$ and $\Lambda_j$ of $G$ such that $\sigma(\Lambda_i) = \Lambda_j$. The downsets (or the edge labelled posets) of $\Lambda_i$ and $\Lambda_j$ must be isomorphic. Therefore, by Proposition \[prop:equiv-elp\], there is a counter example to Ulam’s conjecture.

The proof of [*if*]{} is also done by contradiction. Suppose that Ulam’s conjecture is false, and $G$ and $H$ are connected nonisomorphic graphs having identical vertex decks. We show that $\mathcal{ELP}(G+H)$ has a nontrivial automorphism. Define a bijective map $\sigma : \Lambda(G+H)\rightarrow  \Lambda(G+H)$ as follows.

1.  The graph $G+H$ is mapped to itself.

2.  If $\Lambda_i\in \Lambda(G+H)$ has a component isomorphic to $G$, then denote $\Lambda_i$ by $G+X$, where $X$ is a proper subgraph of the component isomorphic to $H$. In this case, set $\sigma(G+X) = H+X$.

3.  If $\Lambda_i$ is $H+X$, where $X$ is a proper subgraph of the component isomorphic to 