Suppose that we have a non-planar graph $G$ which is undirected and connected. Our aim is to remove a set of edges and/or a set of vertices and convert make $G$ planar while keeping the connectedness.
Besides connectedness, $G$ is guaranteed to have enough connectivity so that removal of a reasonably small amount of vertices do not disconnect the graph and there are several paths between two specified vertices (removing one or two vertices do not let $G$ be a tree).
I have looked it up on the Internet and could not find any research. Maybe it is because I do not know the proper keywords.
Is there any research on this topic? If so, could you provide a reference?