Optimal vertex-disjoint cycle covers of weighted symmetric graphs with $n$ vertices can be calculated efficiently with the method of Tutte.
It is also possible to efficiently calculate optimal matchings with $2k<n$ edges.


  

**Question:**


Is it also possible to reduce the calculation of optimal vertex-disjoint cycle covers with $k<n$ edges to matching?



Finding optimal cycle covers with fixed number of vertices
Just realized that the method for finding optimal matchings of fixed size can easily be generalized to arbitrary $f$-factors.


In order to calculate an optimal $1$-factor whose edges are adjacent to $k<n$ vertices, one adds $n-k$ vertices to the graph, connects each added vertex to all $n$ vertices of the original graph and assigns weight $0$ to the newly added edges.


The generalization to $f$-factors is to add $\ n-k\quad K\_f\ $ to the graph, again connect every of the newly added vertices to each of the $n$ vertices of the original graph and assign weight $0$ to each of the newly added edges.  

As a final step perform the transformation of Tutte or Lovasz and Plummer and calculate the optimal matching in that transformed graph.


