Given an undirected graph $$$G = (V, E)$$$ and $$$u, v \in V$$$. Determine if there exists a sub-graph $$$G' = (V', E') \subseteq G$$$ and two spanning trees $$$T_1 = (V', E_1), T_2 = (V', E_2)$$$ of $$$G'$$$, such that

• $$$u, v \in V'$$$
• $$$E_1 \cap E_2 = \varnothing$$$

It would be better if you can find out such $$$T_1, T_2$$$.