The current chapter is based on many sources; see Section 8.6.1 for details.

Organization. The current chapter is the first of a series of three chapters that cover three models for testing graph properties. In each model, we spell out the definition of property testing (when specialized to that model), present some of the known results, and demonstrate some of the ideas involved in the construction of testers (by focusing on testing Bipartiteness, which seems a good benchmark).

We start the current chapter with a general introduction to testing graph properties, which includes an overview of the three models (see Section 8.1.2). We then present and illustrate the “dense graph model” (Section 8.2), which is the focus of the current chapter. The main two sections (i.e., Sections 8.3 and 8.4) cover the two topics that are mentioned in the foregoing summary: Section 8.3 deals with testing arbitrary graph partition properties, as illustrated by the example of testing Bipartitness. Section 8.4 deals with the connection between property testing in this model and Szemeredi's Regularity Lemma, as illustrated by testing subgraph-freeness. The last two sections (i.e., Sections 8.5 and 8.6) are descriptive in nature: Section 8.5 presents a taxonomy of the known results, whereas Section 8.6 presents final comments.