Not sure how to phrase this really, but here goes.
Suppose you are given $k$ simple graphs, each having exactly $m$ edges. The edges in each graph are labeled from 1 to $m$.
The problem is to find the maximum subset of edges $M \subseteq \{1, 2, ..., m\}$ you can choose, so that $M$ forms a forest in each of the $k$ graphs.
For $k = 1$ this is trivial, but starting from $k = 2$ the problem seems to be harder.
Has this problem been studied extensively?
The problem I am interested in is actually a generalization of the above, where there can be multiple edges between two vertices. (kind of like a weighted version, in some sense)