  We address the induced matching enumeration problem. An edge set $M$ is an
induced matching of a graph $G =(V,E)$. The enumeration of matchings are widely
studied in literature, but the induced matching has not been paid much
attention. A straightforward algorithm takes $O(|V|)$ time for each solution,
that is coming from the time to generate a subproblem. We investigated local
structures that enables us to generate subproblems in short time, and proved
that the time complexity will be $O(1)$ if the input graph is $C_4$-free. A
$C_4$-free graph is a graph any whose subgraph is not a cycle of length four.
Finally, we show the fixed parameter tractability of counting induced matchings
for graphs with bounded tree-width and planar graphs.
