  An independent set $I_c$ is a \textit{critical independent set} if $|I_c| -
|N(I_c)| \geq |J| - |N(J)|$, for any independent set $J$. The \textit{critical
independence number} of a graph is the cardinality of a maximum critical
independent set. This number is a lower bound for the independence number and
can be computed in polynomial-time. Any graph can be decomposed into two
subgraphs where the independence number of one subgraph equals its critical
independence number, where the critical independence number of the other
subgraph is zero, and where the sum of the independence numbers of the
subgraphs is the independence number of the graph. A proof of a conjecture of
Graffiti.pc yields a new characterization of K\"{o}nig-Egervary graphs: these
are exactly the graphs whose independence and critical independence numbers are
equal.
