similar to Kelly’s Lemma  \[lem-kelly\]. Each induced subgraph of $H$ that is isomorphic to $F$ is also an induced subgraph of $\displaystyle\binom{v(H)-v(F)}{q-v(F)}$ induced subgraphs of $H$ that have $q$ vertices.

\[lem-elp\_eq\_n\] The structures $\mathcal{N}(G)$ and $\mathcal{ELP}(G)$ can be constructed from each other.

We first show how $\mathcal{N}(G)$ is constructed from $\mathcal{ELP}(G)$. The matrix $\mathcal{N}(G)$ is an $I\times I$ matrix where $I$ is the number of points in $\mathcal{ELP}(G)$. Without the loss of generality, suppose that the points of $\mathcal{ELP}(G)$ are labelled from $\Lambda_1$ to $\Lambda_I$ such that if $\rho(\Lambda_i) < \rho(\Lambda_j)$ then $i < j$, where $\rho$ is the rank function defined in Lemma \[lem-rank\]. Correspondingly, the rows and the columns of $\mathcal{N}(G)$ are indexed from $\Lambda_1$ to $\Lambda_I$. The edge labels in $\mathcal{ELP}(G)$ immediately give some of the entries in $\mathcal{N}(G)$: if $\Lambda_i $ covers $\Lambda_j$ then $N_{ij}$ is the label on the edge joining $\Lambda_i $ and $\Lambda_j$. The diagonal entries are 1. Except $N_{11}$, all the other entries in the first row are 0. We construct the remaining entries of $\mathcal{N}(G)$ by induction on the rank. The base case is rank 2. It corresponds to the first row, and is already filled. Let $f(r)$ denote the number 