of $2G$. Moreover, $v(X) < v(G)$ and $v(Y) < v(G)$. Now $\displaystyle\binom{2G}{F}=\displaystyle\binom{2H}{F}$ follows from the fact that $G$ and $H$ have identical vertex decks and Kelly’s Lemma.

The following corollary is an immediate consequence of the above lemma.

\[cor-correspondence\] Define a correspondence $f$ between $\Lambda(2G)$ and $\Lambda(2H)$ as follows.

1.  $f(2G) = 2H$

2.  $F\in \Lambda(2G)$ is not $2G$ but has a component isomorphic to $G$. We write $F= G+X$, and set $f(F) = H+X$.

3.  $F\in \Lambda(2G)$ has no component isomorphic to $G$. In this case we set $f(F) = F$.

The correspondence defined above is a bijection.

\[lem-2g2h\] $\mathcal{N}(2G) = \mathcal{N}(2H)$.

For the bijection $f$ between non-empty induced subgraphs of $2G$ and $2H$ that was defined in Corollary \[cor-correspondence\], we show that, for any two nonisomorphic induced subgraphs $F_1$ and $F_2$ of $2G$,

$$\label{eq-2g2h}
\displaystyle\binom{F_2}{F_1} = \displaystyle\binom{f(F_2)}{f(F_1)}$$

In view of Corollary \[cor-correspondence\], it is sufficient to show this when at least one of the graphs $F_1$ and $F_2$ has a component isomorphic to $G$.

1.  When $F_2 = 2G$, then Equation (\[eq-2g2h\]) follows from Lemma \[lem-kelly2\] and Corollary \[cor-correspondence\].

2.  When $F_1 = G+X$ and $F_2 = G+Y$, and $v(X) < v(G)$ and $v(Y) < v(G)$, we have 