Suppose I have a weighted undirected graph $G$. I want to find a set of edges $E_{rem}$ to remove from $G$ and create $G'$ such that
- in every component of $G'$, there is at least one node connected to all others (or the minimum spanning tree of each component contains just one node)
- $\sum_{e \in E_{rem}} w(e)$ is minimal, where $w(e)$ denotes the edge weight
Is there existing work like this / a nice connection to other problems? I can envision some greedy algorithms but am wondering if a deterministic/exhaustive solution exists in reasonable complexity?