  Let $F$ be a connected graph with $\ell$ vertices. The existence of a
subgraph isomorphic to $F$ can be defined in first-order logic with quantifier
depth no better than $\ell$, simply because no first-order formula of smaller
quantifier depth can distinguish between the complete graphs $K_\ell$ and
$K_{\ell-1}$. We show that, for some $F$, the existence of an $F$ subgraph in
\emph{sufficiently large} connected graphs is definable with quantifier depth
$\ell-3$. On the other hand, this is never possible with quantifier depth
better than $\ell/2$. If we, however, consider definitions over connected
graphs with sufficiently large treewidth, the quantifier depth can for some $F$
be arbitrarily small comparing to $\ell$ but never smaller than the treewidth
of $F$. Moreover, the definitions over highly connected graphs require
quantifier depth strictly more than the density of $F$. Finally, we determine
the exact values of these descriptive complexity parameters for all connected
pattern graphs $F$ on 4 vertices.
