be calculated from $\mathcal{VD}(G)$. For example, Kelly’s Lemma allows us to count the number of vertex-proper subgraphs of $G$ of any given type.

\[lem-kelly\][**(Kelly’s Lemma [@kelly1957])**]{} If $F$ is a graph such that $v(F) < v(G)$ then $$\label{eq-kelly}
{\mbox{$\left[\begin{array}{c}G\\#2\end{array}\right]$}} = \frac{1}{v(G)-v(F)}\sum_{u\in VG}{\mbox{$\left[\begin{array}{c}G-u\\#2\end{array}\right]$}}$$ therefore, ${\mbox{$\left[\begin{array}{c}G\\#2\end{array}\right]$}}$ is reconstructible from $\mathcal{VD}(G)$. Also, in Equation (\[eq-kelly\]), ${\mbox{$\left[\begin{array}{c}G\\#2\end{array}\right]$}}$ and ${\mbox{$\left[\begin{array}{c}G-u\\#2\end{array}\right]$}}$ may be replaced by $\displaystyle\binom{G}{F}$ and $\displaystyle\binom{G-u}{F}$, respectively.

Tutte [@tutte1977], [@tutte1984] proved the reconstructibility of the characteristic polynomial and the chromatic polynomial. Tutte’s results were simplified by an elegant counting argument by Kocay [@kocay1981]. This argument is useful to count certain subgraphs that span $VG$.

Let $S = \{F_1, F_2, \ldots ,F_k\}$ be a family of graphs. Let $c(S,H)$ be the number of tuples $(X_1, X_2, \ldots ,X_k)$ of subgraphs of $H$ such that $X_i\cong F_i\, \forall \, i$, and $\cup_{i = 1}^{k}X_i = H$. We call it the number of [*$S$-covers*]{} of $H$.

\[lem-kocay\] [**(Kocay’s Lemma [@kocay1981])**]{} $$\prod_{i = 1}^{k} {\mbox{$\left[\begin{array}{c}G\\#2\end{array}\right]$}} = \sum_{X} c(S,X){\mbox{$\left[\begin{array}{c}G\\#2\end{array}\right]$}}$$ where the summation is over all isomorphism types of subgraphs of $G$. Also, if $v(F_i)\, <\, v(G)\,\forall\, i$ then $ \sum_{X} c(S,X){\mbox{$\left[\begin{array}{c}G\\#2\end{array}\right]$}}$ over all isomorphism types $X$ of spanning subgraphs of $G$ can be reconstructed from the vertex deck of 