Graph Theory - Bipartite Matchings

Question

Under what conditions can the symmetric difference between a (non-maximal) matching $M$ and a maximal matching $M*$ contain a path of length $\geq 2$? The problem is to show that every vertex in the symmetric difference of $M$ and $M*$ has degree $1$ or $2$. The endpoints of a path of any length clearly have degree $1$ and this path(s) always exists, but I'm struggling with showing how a path of length greater than $1$ CAN exist...

Answer 1

Take a cycle of even length, set $M^*$ to contain every second edge and set $M$ to be a complement of $M^*$ minus some one edge. You will get something like below:

I hope this helps $\ddot\smile$