 
Summary: Testing perfection is hard
Noga Alon
Jacob Fox
Abstract
A graph property P is strongly testable if for every fixed > 0 there is a onesided tester for
P whose query complexity is bounded by a function of . In classifying the strongly testable graph
properties, the first author and Shapira showed that any hereditary graph property (such as P the
family of perfect graphs) is strongly testable. A property is easily testable if it is strongly testable
with query complexity bounded by a polynomial function of 1
, and otherwise it is hard. One of
our main results shows that testing perfectness is hard. The proof shows that testing perfectness
is at least as hard as testing trianglefreeness, which is hard. On the other hand, we show that
induced P3freeness 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 Hfreeness is easily testable.
1 Introduction
Property testing is an active area of computer science where one wishes to quickly distinguish between
objects that satisfy a property from objects that are far from satisfying that property. The study
of this notion was initiated by Rubinfield and Sudan [22], and subsequently Goldreich, Goldwasser,
and Ron [14] started the investigation of property testers for combinatorial objects. Graph property
