  A graph property P is strongly testable if for every fixed \epsilon>0 there
is a one-sided \epsilon-tester for P whose query complexity is bounded by a
function of \epsilon. 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 \epsilon^{-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 triangle-freeness, which is hard. On
the other hand, we show that induced P_3-freeness is easily testable. This
settles one of the two exceptional graphs, the other being C_4 (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.
