  Let $G$ be a graph that admits a perfect matching. A {\sf forcing set} for a
perfect matching $M$ of $G$ is a subset $S$ of $M$, such that $S$ is contained
in no other perfect matching of $G$. This notion originally arose in chemistry
in the study of molecular resonance structures. Similar concepts have been
studied for block designs and graph colorings under the name {\sf defining
set}, and for Latin squares under the name {\sf critical set}. Recently several
papers have appeared on the study of forcing sets for other graph theoretic
concepts such as dominating sets, orientations, and geodetics. Whilst there has
been some study of forcing sets of matchings of hexagonal systems in the
context of chemistry, only a few other classes of graphs have been considered.
  Here we study the spectrum of possible forced matching numbers for the grids
$P_m \times P_n$, discuss the concept of a forcing set for some other specific
classes of graphs, and show that the problem of finding the smallest forcing
number of graphs is \NP--complete.
