  In matching theory, barrier sets (also known as Tutte sets) have been studied
extensively due to its connection to maximum matchings in a graph. In this
paper, we first define $\theta$-barrier sets. Our definition of a
$\theta$-barrier set is slightly different from that of a barrier set. However
we show that $\theta$-barrier sets and barrier sets have similar properties. In
particular, we prove a generalized Berge's Formula and give a characterization
for the set of all $\theta$-special vertices in a graph.
