Bump bump ... I've noticed that $477=\frac{1}{2}(2016-1062)$. I want to find a maximal chain $C_1-\ldots-C_k$, then all neighbourhood of $C_k$ are in $\{C_1,\ldots,C_{k-1}\}$. Then we may define $A=V\backslash\{C_1,\ldots,C_k\}$, $B=\{C_k\}$. Then I want to find some vertices in $A$ and change them into $B$. For example if the chain $C_1-\ldots-C_{k-1}$ can be extended to some $C_1-\ldots-C_{k-1}-D$, then we can change $D$ into $B$. But it is not enough. If we extend $C_1-\ldots-C_{k-i}$ to $\ldots-C_{k-i}-D_i-\ldots-D_1$, then we need to change all $D_i,\ldots,D_1$ into $B$, because all neighbourhoods of $D_1$ are in the set $\{C_1,\ldots,C_{k-i},D_i,\ldots,D_2\}$. Those elements cannot stay in $A$. So if we want to change elements from $A$ into $B$, it's hard to explain why we can make $A,B$ have exactly $477$ elements?