  We consider algorithms for finding and counting small, fixed graphs in sparse
host graphs. In the non-sparse setting, the parameters treedepth and treewidth
play a crucial role in fast, constant-space and polynomial-space algorithms
respectively. We discover two new parameters that we call matched treedepth and
matched treewidth. We show that finding and counting patterns with low matched
treedepth and low matched treewidth can be done asymptotically faster than the
existing algorithms when the host graphs are sparse for many patterns. As an
application to finding and counting fixed-size patterns, we discover
$\otilde(m^3)$-time \footnote{$\otilde$ hides factors that are logarithmic in
the input size.}, constant-space algorithms for cycles of length at most $11$
and $\otilde(m^2)$-time, polynomial-space algorithms for paths of length at
most $10$.
