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].

翻译：我们重新审视了[Atserias等，JACM 2006]和[Dawar，JCSS 2010]开创的关于稀疏结构类中一阶逻辑同态保持的研究工作。这些工作确立了一阶逻辑在任意单调且可加稀疏类中具有同态保持性质。然而，可加性假设对于所给出的证明而言并不足够充分。我们通过构造具有有界树宽、单调且可加但缺乏同态保持性质的图类来证明这一点。我们还证明了同态保持性质在平面图类上不成立。另一方面，通过用瓶颈上的融合这一更强条件替代可加性，可以恢复同态保持的证明。这与[Atserias等，SiCOMP 2008]中为扩展保持所提出的类似条件具有类比性。