Mutually exclusive replacement paths, an existing problem? A replacement path is a simple path allocated to an edge $e \in G$
that connects the endpoints of $e$ in $G \setminus \{e\}$.

The problem

An undirected graph $G$ is given and the task is to allocate a replacement path $P_e$ to each edge 
$e \in G$ such that for any pair of edges $e_1$ and $e_2$ it holds:
if $e_2 \in P_{e_1}$ then $e_1 \not\in P_{e_2}$.
In words, no two edges use each other in their replacement path.
I call the paths $P_{e_1}$ and $P_{e_2}$ mutually exclusive.

The path length is not bounded and they can share edges. So any path allocation that satisfies the
mutual exclusion would be good.

Now the question: is there already any problem closely related to this?

One could reduce this problem to Maximum Independent Sets problem: a vertex for every $(e,P_e)$ and edges connecting conflicting pairs (i.e. pairs violating the ME). However, in order to get a polynomial size instance we need to restrict the model so that each edge gets a constant number of candidate paths.
However, this is not desired because the MIS instances seem to be NP-Hard (or not?).
I found a replacement path allocation for $_$ in a more general setting. Consider an edge $_1$ that uses edge $_2$ on its path. The allocation induces a dependency from $_1$ to $_2$. Mutual exclusion defined above corresponds to a cycle of 2 such dependencies. Now consider the more general setting: a cycle of $$ dependencies induced by a path allocation on $_$. Assume vertices are $0,…,−1$. On edges $,$, the path allocation $,+1,0,+1,$ induces cycles of dependencies no shorter than $$. In my application it means that $_$ can tolerate up to $−1$ edge failures (i.e. edge removal).

Same problem on hypercubes, published here:
''Local Fast Segment Rerouting on Hypercubes''

Further solutions were submitted recently. Will update soon.