We consider the notion of a local-characterization of an infinite family of unlabeled bounded-degree graphs. Such a local-characterization is defined in terms of a finite set of (marked) graphs yielding a generalized notion of subgraph-freeness, which extends the standard notions of induced and non-induced subgraph freeness.
We survey the work of Adler, Kohler and Peng (32nd SODA and 36th CCC, 2021), which is pivoted at constructing locally-characterized expander graphs. The construction makes inherent use of the iterative and local nature of the Zig-Zag construction of Reingold, Vadhan, and Wigderson (41st FOCS, 2000). This yields a locally-characterizable graph property that cannot be tested (in the bounded-degree graph model) within a number of queries that does not depend on the size of the graph.