 
Summary: A Characterization of the (natural) Graph Properties Testable with
OneSided Error
Noga Alon
Asaf Shapira
Abstract
The problem of characterizing all the testable graph properties is considered by many to be
the most important open problem in the area of propertytesting. 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 onesided error tester, if and only if P is semi hereditary. We stress that
any "natural" property that can be tested (either with onesided or with twosided 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 onesided error.
One of the main technical contributions of this paper is in showing that any hereditary graph
property can be tested with onesided error. This general result contains as a special case all the
previous results about testing graph properties with onesided error. These include the results
of [23] and [6] about testing kcolorability, the characterization of [25] of the graphpartitioning
problems that are testable with onesided error, the induced vertex colorability properties of [4],
the induced edge colorability properties of [17], a transformation from twosided to onesided
