Summary: A Characterization of the (natural) Graph Properties Testable with
The problem of characterizing all the testable graph properties is considered by many to be
the most important open problem in the area of property-testing. Our main result in this paper
is a solution of an important special case of this general problem; Call a property tester oblivious
if its decisions are independent of the size of the input graph. We show that a graph property P
has an oblivious one-sided error tester, if and only if P is (almost) hereditary. We stress that
any "natural" property that can be tested (either with one-sided or with two-sided error) can be
tested by an oblivious tester. In particular, all the testers studied thus far in the literature were
oblivious. Our main result can thus be considered as a precise characterization of the "natural"
graph properties, which are testable with one-sided error.
One of the main technical contributions of this paper is in showing that any hereditary graph
property can be tested with one-sided error. This general result contains as a special case all the
previous results about testing graph properties with one-sided error. These include the results
of  and  about testing k-colorability, the characterization of  of the graph-partitioning
problems that are testable with one-sided error, the induced vertex colorability properties of ,
the induced edge colorability properties of , a transformation from two-sided to one-sided