The current chapter is based on several sources; see Section 10.5.2 for details.

Organization. Following an introduction to the general graph model (Section 10.1), we study the issues that arise when trying to extend testers for the bounded-degree graph model to testers for the current model (Section 10.2). Next, in Section 10.3, we study the related problems of estimating the average degree in a general graph and selecting random edges in it, presenting two different algorithmic approaches toward solving these problems (see Sections 10.3.2.1 and 10.3.2.2, respectively). As illustrated in Section 10.2.2, these problems are pivotal for the design of some testers. Lastly, in Section 10.4, we illustrate the possible benefits of using both incidence and gadjacency queries.

The General Graph Model: Definitions and Issues

The general graph model is intended to capture arbitrary graphs, which may be neither dense nor of bounded degree. Such graphs occur most naturally in many settings, but they are not captured (or not captured well) by the models presented in the previous two chapters (i.e., the dense graph model and the bounded-degree graph model).

Recall that both in the dense graph model and in the bounded-degree graph model, the query types (i.e., ways of probing the tested graph) and the distance measure (i.e., distance between graphs) were linked to the representation of graphs as functions. In contrast to these two models, in the general graph model the representation is blurred, and the query types and distance measure are decoupled.