we assume it to be the first row. Now $e(\Lambda_i) = N_{i1}$ for each $\Lambda_i$.

By Lemma \[lem-elp\_eq\_n\], $\mathcal{ELP}(G)$ is uniquely constructed. By Lemma \[lem-rank\], the rank function of the poset defined by $\rho(\Lambda_1)=2$ gives $v(\Lambda_i)=\rho(\Lambda_i)$ for each $\Lambda_i$.

Now on, without the loss of generality, we will assume that the nonisomorphic induced subgraphs $\Lambda_1, \Lambda_2, \ldots,
\Lambda_I$ of a graph $G$ under consideration are ordered so that $v(\Lambda_i)$ are in a non-decreasing order. The first row will correspond to $\Lambda_1 \cong K_2$ and the last row to $\Lambda_I \cong G$.

\[lem-kelly3\] The collection $\{\mathcal{N}(G-u)| u \in VG, e(G-u) > 0\}$ is unambiguously determined by $\mathcal{N}(G)$.

Note that this collection is a “multiset”, that is, an $\mathcal{N}$-matrix may appear multiple times in the collection.

Let $j \neq I$. The graph $\Lambda_j$ is a vertex deleted subgraph of $G$ if and only if for all $i \neq j \neq I$, $N_{ij}=0$. Now $N(\Lambda_j)$ is obtained by deleting $k$’th row and $k$’th column for each $k$ such that $N_{jk} = 0$. A multiplicity $N_{Ij}$ is assigned to $N(\Lambda_j)$. Equivalently, we can construct $\mathcal{ELP}(G)$ by Lemma \[lem-elp\_eq\_n\], then construct the down set $\mathcal{ELP}(\Lambda_j)$ of each $\Lambda_j$ that is covered by $\Lambda_I = G$, 