  We revisit the relation between two fundamental property testing models for
bounded-degree directed graphs: the bidirectional model in which the algorithms
are allowed to query both the outgoing edges and incoming edges of a vertex,
and the unidirectional model in which only queries to the outgoing edges are
allowed. Czumaj, Peng and Sohler [STOC 2016] showed that for directed graphs
with both maximum indegree and maximum outdegree upper bounded by $d$, any
property that can be tested with query complexity $O_{\varepsilon,d}(1)$ in the
bidirectional model can be tested with $n^{1-\Omega_{\varepsilon,d}(1)}$
queries in the unidirectional model. In particular, if the proximity parameter
$\varepsilon$ approaches $0$, then the query complexity of the transformed
tester in the unidirectional model approaches $n$. It was left open if this
transformation can be further improved or there exists any property that
exhibits such an extreme separation.
  We prove that testing subgraph-freeness in which the subgraph contains $k$
source components, requires $\Omega(n^{1-\frac{1}{k}})$ queries in the
unidirectional model. This directly gives the first explicit properties that
exhibit an $O_{\varepsilon,d}(1)$ vs $\Omega(n^{1-f(\varepsilon,d)})$
separation of the query complexities between the bidirectional model and
unidirectional model, where $f(\varepsilon,d)$ is a function that approaches
$0$ as $\varepsilon$ approaches $0$. Furthermore, our lower bound also resolves
a conjecture by Hellweg and Sohler [ESA 2012] on the query complexity of
testing $k$-star-freeness.
