  We show that bordered Heegaard Floer homology detects incompressible surfaces
and bordered-sutured Floer homology detects partly boundary parallel tangles
and bridges, in natural ways. For example, there is a bimodule Lambda so that
the tensor product of CFD(Y) and Lambda is Hom-orthogonal to CFD(Y) if and only
if the boundary of Y admits an essential compressing disk. In the process, we
sharpen a nonvanishing result of Ni's. We also extend
Lipshitz-Ozsv\'ath-Thurston's "factoring" algorithm for computing HF-hat to
compute bordered-sutured Floer homology, to make both results on detecting
incompressibility practical. In particular, this makes Zarev's tangle invariant
manifestly combinatorial.
