  We revisit the work studying homomorphism preservation for first-order logic
in sparse classes of structures initiated in [Atserias et al., JACM 2006] and
[Dawar, JCSS 2010]. These established that first-order logic has the
homomorphism preservation property in any sparse class that is monotone and
addable. It turns out that the assumption of addability is not strong enough
for the proofs given. We demonstrate this by constructing classes of graphs of
bounded treewidth which are monotone and addable but fail to have homomorphism
preservation. We also show that homomorphism preservation fails on the class of
planar graphs. On the other hand, the proofs of homomorphism preservation can
be recovered by replacing addability by a stronger condition of amalgamation
over bottlenecks. This is analogous to a similar condition formulated for
extension preservation in [Atserias et al., SiCOMP 2008].
