Some structural relationships between matchings and independent sets a
re presented. One consequence is that any NC algorithm for generating
a uniform random perfect matching of a bipartite graph must differ fro
m the only known polynomial time random matching algorithm, due to Bro
der (1986) and Jerrum and Sinclair (1988), unless NC = P. Efficient pa
rallel algorithms for emulating random walk on general graphs and some
implicitly represented large-scale graphs are also presented.