A Characterization of the (Natural) Graph Properties Testable with One-Sided Error

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 property-testing. 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 one-sided error tester, if and only if P is (almost) hereditary. We stress that any "natural" property that can be tested (either with one-sided or with two-sided 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 one-sided error.One of the main technical contributions of this paper is in showing that any hereditary graph property can be tested with one-sided error. This general result contains as a special case all the previous results about testing graph properties with one-sided error. These include the results of [20] and [5] about testing k-colorability, the characterization of [21] of the graph-partitioning problems that are testable with one-sided error, the induced vertex colorability properties of [3], the induced edge colorability properties of [14], a transformation from two-sided to one-sided error testing [21], as well as a recent result about testing monotone graph properties [10]. More importantly, as a special case of our main result, we infer that some of the most well studied graph properties, both in graph theory and computer science, are testable with one-sided error. Some of these properties are the well known graph properties of being Perfect, Chordal, Interval, Comparability, Permutation and more. None of these properties was previously known to be testable.

Distribution-Free Connectivity Testing for Sparse Graphs

“…Among these problems are k-colorability for any fixed k ≥ 2, ρ-clique for any ρ > 0 (testing whether there is a clique in G of size at least ρ · n), ρ-cut for any ρ > 0 (testing whether there is a cut in G containing at least ρ · n 2 edges) and ρ-bisection. Testing in this graph model was further studied, for example in [2][3][4][5][6].…”

“…3 It is not known whether membership queries are essential in general for testing even in the uniform case (see [39]); such a result is known only for specific problems such as monotonicity testing (see [17]). 4 In order to allow parallel edges in the definition, we refer to the j th copy of the edge (u, v) as a distinct edge (u j , v j ). G are C 1 , .…”

Generalizations of the removal lemma

“…In particular, the result of Alon and Shapira in [2] is a generalization, which extends all the previous results of this type, where the triangle is replaced by a possibly infinite family of graphs and containment is induced. The main result of the present paper is Theorem 6, which is an extension of the result of Alon and Shapira from graphs to k-uniform hypergraphs.…”

“…More general statements of that type regarding graphs were successively proved by several authors in [1][2][3]10]. In particular, the result of Alon and Shapira in [2] is a generalization, which extends all the previous results of this type, where the triangle is replaced by a possibly infinite family of graphs and containment is induced.…”

“…In [2] Alon and Shapira proved a common generalization of Theorem 3 and Theorem 4, extending Theorem 4 from one forbidden induced graph F to a forbidden family of induced graphs F . The aim of this paper is to extend their result to k-uniform hypergraphs.…”