Testing perfection is hard

A graph property P is strongly testable if for every fixed ε>0 there is a one-sided ε-tester for P whose query complexity is bounded by a function of ε. 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 ε^{-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.

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