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Two independent spanning trees of $2$-connected graph

I want to prove the following statement: Let $u$ be a vertex in a $2$-connected graph $G$. Then $G$ has two spanning trees such that for every vertex $v$, the $u,v$-paths in the trees are independent …
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Two independent spanning trees of $2$-connected graph

Let's use a new concept oriented path in a cycle $C$ or an open ear $P_k$. In $C$: Label the vertices in $C$ in clockwise direction by $u,v_1,\cdots,v_n$. Then the clockwise-oriented path of $C$ ($c_ …
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