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.

A common thread in all the recent results concerning testing dense graphs is the use of Szemerédi's regularity lemma. In this paper we show that in some sense this is not a coincidence. Our first result is that the property defined by having any given Szemerédi-partition is testable with a constant number of queries. Our second and main result is a purely combinatorial characterization of the graph properties that are testable with a constant number of queries. This characterization (roughly) says that a graph property P can be tested with a constant number of queries if and only if testing P can be reduced to testing the property of satisfying one of finitely many Szemerédi-partitions. This means that in some sense, testing for Szemerédi-partitions is as hard as testing any testable graph property. We thus resolve one of the main open problems in the area of property-testing, which was first raised in the 1996 paper of Goldreich, Goldwasser and Ron [24] that initiated the study of graph property-testing. This characterization also gives an intuitive explanation as to what makes a graph property testable. Basic definitionsThe meta problem in the area of property testing is the following: Given a combinatorial structure S, distinguish between the case that S satisfies some property P and the case that S is -far from satisfying P. Roughly speaking, a combinatorial structure is said to be -far from satisfying some property P if an -fraction of its representation should be modified in order to make S satisfy P. The main goal is to design randomized algorithms, which look at a very small portion of the input, and using this information distinguish with high probability between the above two cases. Such algorithms are called property testers or simply testers for the property P. Preferably, a tester should look at a portion of the input whose size is a function of only. Blum, Luby and Rubinfeld [10] were the first to formulate a question of this type, and the general notion of property testing was first formulated by Rubinfeld and Sudan [34], who were interested in studying various algebraic properties such as linearity of functions.The main focus of this paper is the testing of properties of graphs. More specifically, we focus on property testing in the dense graph model as defined in [24]. In this case a graph G is said to be -far from satisfying a property P, if one needs to add/delete at least n 2 edges to G in order to turn it into a graph satisfying P. A tester for P should distinguish with high probability, say 2/3, between the case that G satisfies P and the case that G is -far from satisfying P. Here we assume that the tester can query some oracle whether a pair of vertices, i and j, are adjacent in the input graph G. In what follows we will say that a tester for a graph property P has one-sided error if it accepts any graph satisfying P with probability 1 (and still rejects those that are -far with probability at least 2/3). If the tester may reject graphs satisfying P with non-zero probability then...

The present paper is based on 378 A-p elastic scattering events in the incident-momentum region 120-320 MeV/c. Differential and total cross sections have been measured in several momentum intervals and found to be consistent with predominantly S-wave scattering. No significant indication for the existence of a low-energy A-p resonance has been found. Using the effective-range approximation, the four scattering parameters a s , a t , r Sf and r t were evaluated with and without further assumptions on the A-p interaction properties. Best values obtained from the four-parameter fit were #«= -1.8, a t = -1.6, r s = 2.8, and r t = 3.3 F. A likelihood-function mapping procedure is used to describe the large and strongly correlated errors of these values. LOW-ENERGYA-p ELASTIC SCATTERING

Suppose G is a graph of bounded degree d, and one needs to remove n of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G. In fact, a similar result is proved for any minor-closed property of bounded degree graphs. The main motivation of the above result comes from theoretical computer-science. Using our main result we infer that for any minor-closed property P, there is a constant time algorithm for detecting if a graph is "far" from satisfying P. This, in particular, answers an open problem of Goldreich and Ron [STOC 1997], who asked if such an algorithm exists when P is the graph property of being planar. The proof combines results from the theory of graph minors with results on convergent sequences of sparse graphs, which rely on martingale arguments.

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.