Beyond Expectations

You are given an undirected simple graph $G$ with the set of its vertices $V$, consisting of $N$ vertices (numbered $1$ to $N$) and the set of edges $E$, having $M$ edges (numbered $1$ to $M$). Each edge $i$ has $2$ lists of vertices ($A_i$ and $B_i$) associated with it. These $2$ lists are not necessarily disjoint. For every subset $V_1$ of $V$, we define the special graph $G_1 ( V_1, E_1 )