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 [Ateserias et al., SiCOMP 2008].

翻译：本文重新审视了由[Atserias 等, JACM 2006]和[Dawar, JCSS 2010]开创的关于一阶逻辑在稀疏结构类中同态保持性质的研究。这些研究曾确立：在任意单调且可加（addable）的稀疏类中，一阶逻辑具有同态保持性质。然而，可加性这一假设的强度不足以支撑其给出的证明。我们通过构造有界树宽的图类来证明这一点——这些图类是单调且可加的，却缺乏同态保持性质。我们还证明了同态保持性质在平面图类上失效。另一方面，通过用瓶颈上的融合（amalgamation over bottlenecks）这一更强条件替代可加性，可以恢复同态保持性质的证明。这一条件与[Atserias 等, SiCOMP 2008]中为扩展保持性质所表述的类似条件相类似。