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A Characterization of Easily Testable Induced Subgraphs Asaf Shapira

Summary: A Characterization of Easily Testable Induced Subgraphs
Noga Alon
Asaf Shapira
Let H be a fixed graph on h vertices. We say that a graph G is induced H-free if it does not
contain any induced copy of H. Let G be a graph on n vertices and suppose that at least n2
edges have to be added to or removed from it in order to make it induced H-free. It was shown
in [5] that in this case G contains at least f( , h)nh
induced copies of H, where 1/f( , h) is an
extremely fast growing function in 1/ , that is independent of n. As a consequence, it follows that
for every H, testing induced H-freeness with one-sided error has query complexity independent
of n. A natural question, raised by the first author in [1], is to decide for which graphs H the
function 1/f( , H) can be bounded from above by a polynomial in 1/ . An equivalent question
is for which graphs H, can one design a one-sided error property tester for testing induced H-
freeness, whose query complexity is polynomial in 1/ . We settle this question almost completely
by showing that, quite surprisingly, for any graph other than the paths of lengths 1,2 and 3, the
cycle of length 4, and their complements, no such property tester exists. We further show that
a similar result also applies to the case of directed graphs, thus answering a question raised by
the authors in [9]. We finally show that the same results hold even in the case of two-sided error
property testers. The proofs combine combinatorial, graph theoretic and probabilistic arguments


Source: Alon, Noga - School of Mathematical Sciences, Tel Aviv University


Collections: Mathematics