Easily Testable Graph Properties

A graph on n vertices is -far from a property P if one has to add or delete from it at least n edges to get a graph satisfying P. A graph property P is strongly testable (in the dense model) if for every fixed > 0 it is possible to distinguish, with one-sided error, between graphs satisfying P and ones that are -far from P by inspecting the induced subgraph on a random subset of at most f( ) vertices. A property is easily testable if it is strongly testable and the function f is polynomial in 1/ , otherwise it is hard. In classifying the strongly testable graph properties, the first author and Shapira showed that any hereditary graph property is strongly testable and the converse is essentially true as well (for “natural” properties). We consider the problem of characterizing the easily testable graph properties, which is wide open, and obtain several results in its study. One of our main results shows that testing perfectness is hard. The proof shows that testing perfectness is at least as hard as testing triangle-freeness, which is hard. On the other hand, we show that being a cograph, or equivalently, induced P3-freeness where P3 is a path with 3 edges, is easily testable. This settles one of the two exceptional graphs, the other being C4 (and its complement), left open in the characterization by the first author and Shapira of graphs H for which induced H-freeness is easily testable. Our techniques yield a few additional related results, but the problem of characterizing all easily testable graph properties, or even that of formulating a plausible conjectured characterization, remains open.