maximum matching number decreases when vertices collapse? It seems that the maximum matching number decreases when some independent (not connected) vertices collapse into one vertex, but I don't know if it is absolutely true. 
Would maximum matching number increases when two vertices are connected to form a new edge?

Any proof? Thanks!
 
Definitions:
Given a graph G = (V,E), a matching M in G is a set of pairwise non-adjacent edges; that is, no two edges share a common vertex.

A maximum matching (also known as maximum-cardinality matching1) is a matching that contains the largest possible number of edges.
Try to find a proof that it can't happen! Keep trying.  Hint:


   Suppose you have a maximum matching for the collapsed graph.  Can you use that to find a matching for the original graph?  How many edges does it have?  What does that imply about your question?


If you're really stuck, try to write a program to enumerate all graphs with at most 6 vertices (say) and see if the matching number can ever increase after collapsing an edge.  That should enable you to form a reasonable conjecture, which you can then try to prove, or should give you a counterexample.