I am trying to find polynomial, indicator function or sometimes called structure function to express whether a vertex-induced random subgraph $H$ of $G$ is connected or not. The polynomial $\phi(G')$ should be the undirected version while I am trying to figure out the direction version. So
where I cannot fully understand the second line: it multiplies all $p_i$ of $V(H)$ and then multiplies the probabilities of the rest vertices. What does this tell about connectivity?
What are the connectivity polynomials (also called indicator function or structure function) for the vertex-induced random subgraphs of $G$?