I asked this question on Math Stackexchange but it didn't get much attention, so I am asking it here.
Edmond's Blossom algorithm (Wikipedia), or simply the blossom algorithm, is a popular graph algorithm to construct a maximum matching in a graph (matching is a set of edges without any common vertex; maximum matching is a matching with the highest cardinality of this set).
I do understand the idea behind finding augmenting paths and reversing the matching on them to get a matching with one higher cardinality,
what I don't understand is why odd cycles are a problem in finding an augmenting path (the algorithm says to "shrink" any odd cycle found while searching for an augmenting path).
My best guess is that an odd cycle doesn't allow you to visit some path connected to the cycle because you come back along the same path that led you to the cycle? but I am not sure.
How does an odd cycle create a problem, but even cycle doesn't?
A simple visual description, if someone can provide, might also be helpful.
Thanks in advance