Watermarking is a way of embedding information in digital documents. Much research has been done on techniques for watermarking relational databases and XML documents, where the process of embedding information shouldn't distort query outputs too much. Recently, techniques have been proposed to watermark some classes of relational structures preserving first-order and monadic second order queries. For relational structures whose Gaifman graphs have bounded degree, watermarking can be done preserving first-order queries.

We extend this line of work and study watermarking schemes for other classes of structures. We prove that for relational structures whose Gaifman graphs belong to a class of graphs that have locally bounded tree-width and is closed under minors, watermarking schemes exist that preserve first-order queries. We use previously known properties of logical formulas and graphs, and build on them with some technical work to make them work in our context. This constitutes a part of the first steps to understand the extent to which techniques from algorithm design and computational learning theory can be adapted for watermarking.