Groves

We have been mainly working with groves that correspond to perfect matchings of crosses and wrenches graphs.

Assumption: For any perfect matching of a nice graph we can get any other perfect matching of that graph by taking a sequence of rotations of the matching around a face.

Using the above assumption, we wanted to see how a rotation of a face affected the grove that it corresponded to. We have an idea about what it does to the grove. To get the grove that differs from the given grove by a rotation of the matching, you do the following two steps:

  1. Draw a horizontal line through the vertex of the grove corresponding to the face that we are rotating around.
  2. Change the edges of the grove inside the two rhombi adjascent to the vertex that the line crosses.
Picture of Local Moves.
So now we have two questions:

Gabriel's Groves Code
Gabriel's Graphs Code

We now have a more general move that moves between groves and preserves "groviness": For any vertex of degree 1, choose any adjascent rhombus whose edge is not connected to the vertex, and change it. Also change the edge that was originally connected to the vertex.

This move seems to allow us to get from any grove to any other grove.