Complexity of two perfect matchings with minimum shared edges? Perfect Matching problem is polynomial time solvable in general graphs.
Given undirected simple graph, 


  Is the problem of finding two perfect matching with minimum shared edges between them efficiently solvable?


Shared edge is an edge that occurs in both perfect matchings.
Consider a 3-regular graph $G = (V,E)$.

If there is an edge colouring of $G$ with 3 colours, then you have a partition of $E$ in 3 disjoint perfect matchings. In particular, you can find 2 disjoint perfect matchings.

Conversely, if you can find two disjoint perfect matchings $M_1$ and $M_2$, you can also find a third disjoint perfect matching $M_3 = E \setminus (M_1 \cup M_2)$. The partition $\{M_1, M_2, M_3\}$ of $E$ is an edge colouring of $G$ with 3 colours.

Deciding if a 3-regular graph has an edge colouring with 3 colours is NP-hard (Holyer 1981). Hence it is also NP-hard to decide if there are two disjoint perfect matchings (i.e., to decide if the minimum number of shared edges is 0).