$\mathit{GNI} \in \mathrm{PCP}(\mathit{poly}(n),1)$
GNI is the language of nonisomorphic graphs. Given two grapsh $G_0$ and $G_1$ with $n$ vertices, a verifier expects $\pi$ to contain, for each labeled graph $H$ with $n$ vertices, a bit $\pi[H] \in \{0,1\}$ corresponding to whether $H \equiv G_0$ or $H \equiv G_1$ (arbitrary if neither case holds). Then the verifier can pick a random bit $b \in \{0,1\}$ and a random permutation of $G_b$, $H$. The verifier accepts iff the corresponding bit of $\pi[H]$ is $b$. If $G_0 \not\equiv G_1$ then the verifier accepts with probability $1$ while if $G_0 \equiv G_1$, then the probability of accepting is at most $1/2$.
I was reading this slide and got confused about the following part:
A bit $\pi[H]\in\{0,1\}$ corresponding to whether $H=G_0$ or $H=G_1$ (arbitrary if neither case holds)
But what if both cases hold? In that case, $G_0$ and $G_1$ are isomorphic, and how should we assign the bit? I hope that there's an example.