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.