Consider a simple undirected, unweigthed graph $G=(V,E)$ and its edge-induced subgraph $G^{*}=(V^{*},E^{*})$. Suppose that $G$ and $G^{*}$ have $c$ and $c^{*}$ connected components, respectively. Is it true that exactly
$$|V^{*}|-\min\{0,c-c^{*}\}$$
edges can be selected for any spanning tree of $G$ from $E^{*}$? Could somebody give a proof/counterexample? Thank you for the help!
No; for a counterexample let $E^*$ consist of one edge of $G$, and pick any spanning tree.
$\min\{0,c-c^{*}\}=0$ because $c^*=1$ because $G^*$ is connected, so the expression is just $|V^{*}|=2>1$, so it cannot be the number of edges in the spanning tree from $E^*$.