Consider a connected graph $G$ and a connected graph $H$. Their graph ideals are their path ideals. The Alexander duality of the graph ideals give the cut ideals. $G$ and $H$ are not connected to each other by any path between them. Consider that I connect them with an edge $(u,v)$ from $u\in V(G)$ and $v\in V(H)$.
What is the cut ideal of the new connected graph with $G$ and $H$ as subgraphs?
