maximum matching number decreases when vertices collapse?
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.