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16
Every monotone graph property is testable
 Proc. of STOC 2005
, 2005
"... A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most wellstudied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper i ..."
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A graph property is called monotone if it is closed under removal of edges and vertices. Many monotone graph properties are some of the most wellstudied properties in graph theory, and the abstract family of all monotone graph properties was also extensively studied. Our main result in this paper is that any monotone graph property can be tested with onesided error, and with query complexity depending only on ɛ. This result unifies several previous results in the area of property testing, and also implies the testability of wellstudied graph properties that were previously not known to be testable. At the heart of the proof is an application of a variant of Szemerédi’s Regularity Lemma. The main ideas behind this application may be useful in characterizing all testable graph properties, and in generally studying graph property testing. As a byproduct of our techniques we also obtain additional results in graph theory and property testing, which are of independent interest. One of these results is that the query complexity of testing testable graph properties with onesided error may be arbitrarily large. Another result, which significantly extends previous results in extremal graphtheory, is that for any monotone graph property P, any graph that is ɛfar from satisfying P, contains a subgraph of size depending on ɛ only, which does not satisfy P. Finally, we prove the following compactness statement: If a graph G is ɛfar from satisfying a (possibly infinite) set of monotone graph properties P, then it is at least δP(ɛ)far from satisfying one of the properties.
What is the furthest graph from a hereditary property?
, 2006
"... For a graph property P, the edit distance of a graph G from P, denoted EP(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible e ..."
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Cited by 14 (4 self)
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For a graph property P, the edit distance of a graph G from P, denoted EP(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the furthest graph on n vertices from P and what is the largest possible edit distance from P? Denote this maximal distance by ed(n, P). This question is motivated by algorithmic edgemodification problems, in which one wishes to find or approximate the value of EP(G) given an input graph G. A monotone graph property is closed under removal of edges and vertices. Trivially, for any monotone property, the largest edit distance is attained by a complete graph. We show that this is a simple instance of a much broader phenomenon. A hereditary graph property is closed under removal of vertices. We prove that for any hereditary graph property P, a random graph with an edge density that depends on P essentially achieves the maximal distance from P, that is: ed(n, P) = EP(G(n, p(P))) + o(n 2) with high probability. The proofs combine several tools, including strengthened versions of the Szemerédi Regularity Lemma, properties of random graphs and probabilistic arguments. 1
Property testing in hypergraphs and the removal lemma (Extended Abstract)
, 2006
"... Property testers are efficient, randomized algorithms which recognize if an input graph (or other combinatorial structure) satisfies a given property or if it is “far” from exhibiting it. Generalizing several earlier results, Alon and Shapira showed that hereditary graph properties are testable (wit ..."
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Cited by 11 (0 self)
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Property testers are efficient, randomized algorithms which recognize if an input graph (or other combinatorial structure) satisfies a given property or if it is “far” from exhibiting it. Generalizing several earlier results, Alon and Shapira showed that hereditary graph properties are testable (with onesided error). In this paper we prove the analogous result for hypergraphs. This result is an immediate consequence of a (hyper)graph theoretic statement, which is an extension of the socalled removal lemma. The proof of this generalization relies on the regularity method for hypergraphs.
Homomorphisms in graph property testing  A survey
 Electronic Colloquium on Computational Complexity (ECCC), Report
, 2005
"... on the occasion of his 60 th birthday ..."
Extremal results in sparse pseudorandom graphs
 Adv. Math. 256 (2014), 206–290. arXiv:1204.6645 doi:10.1016/j.aim.2013.12.004 MR3177293
"... Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extendin ..."
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Cited by 10 (8 self)
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Szemerédi’s regularity lemma is a fundamental tool in extremal combinatorics. However, the original version is only helpful in studying dense graphs. In the 1990s, Kohayakawa and Rödl proved an analogue of Szemerédi’s regularity lemma for sparse graphs as part of a general program toward extending extremal results to sparse graphs. Many of the key applications of Szemerédi’s regularity lemma use an associated counting lemma. In order to prove extensions of these results which also apply to sparse graphs, it remained a wellknown open problem to prove a counting lemma in sparse graphs. The main advance of this paper lies in a new counting lemma, proved following the functional approach of Gowers, which complements the sparse regularity lemma of Kohayakawa and Rödl, allowing us to count small graphs in regular subgraphs of a sufficiently pseudorandom graph. We use this to prove sparse extensions of several wellknown combinatorial theorems, including the removal lemmas for graphs and groups, the ErdősStoneSimonovits theorem and Ramsey’s
The maximum edit distance from hereditary graph properties
 Journal of Combinatorial Theory, Ser. B
"... For a graph property P, the edit distance of a graph G from P, denoted EP(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the largest possible edit distance of a graph on n vertices from P? Denote ..."
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Cited by 10 (3 self)
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For a graph property P, the edit distance of a graph G from P, denoted EP(G), is the minimum number of edge modifications (additions or deletions) one needs to apply to G in order to turn it into a graph satisfying P. What is the largest possible edit distance of a graph on n vertices from P? Denote this distance by ed(n, P). A graph property is hereditary if it is closed under removal of vertices. In [7], the authors show that for any hereditary property, a random graph G(n, p(P)) essentially achieves the maximal distance from P, proving: ed(n, P) = EP(G(n, p(P))) + o(n 2) with high probability. The proof implicitly asserts the existence of such p(P), but it does not supply a general tool for determining its value or the edit distance. In this paper, we determine the values of p(P) and ed(n, P) for some subfamilies of hereditary properties including sparse hereditary properties, complement invariant properties, (r, s)colorability and more. We provide methods for analyzing the maximum edit distance from the graph properties of being induced Hfree for some graphs H, and use it to show that in some natural cases G(n, 1/2) is not the furthest graph. Throughout the paper, the various tools let us deduce the asymptotic maximum edit distance from some well studied hereditary graph properties, such as being Perfect, Chordal, Interval, Permutation, ClawFree, Cograph and more. We also determine the edit distance of G(n, 1/2) from any hereditary property, and investigate the behavior of EP(G(n, p)) as a function of p. The proofs combine several tools in Extremal Graph Theory, including strengthened versions
An algorithmic version of the hypergraph regularity method (extended abstract
 Proceedings of the IEEE Symposium on Foundations of Computer Science
, 2005
"... Abstract. Extending the Szemerédi Regularity Lemma for graphs, P. Frankl and V. Rödl [14] established a 3graph Regularity Lemma guaranteeing that all large triple systems Gn admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applicati ..."
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Cited by 9 (6 self)
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Abstract. Extending the Szemerédi Regularity Lemma for graphs, P. Frankl and V. Rödl [14] established a 3graph Regularity Lemma guaranteeing that all large triple systems Gn admit bounded partitions of their edge sets, most classes of which consist of regularly distributed triples. Many applications of this lemma require a companion Counting Lemma [30], allowing one to find and enumerate subhypergraphs of a given isomorphism type in a “dense and regular ” environment created by the 3graph Regularity Lemma. Combined applications of these lemmas are known as the 3graph Regularity Method. In this paper, we provide an algorithmic version of the 3graph Regularity Lemma which, as we show, is compatible with a Counting Lemma. We also discuss some applications. 1.
Parameterized complexity of eulerian deletion problems
 Algorithmica
"... Abstract. We study a family of problems where the goal is to make a graph Eulerian by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc ..."
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Abstract. We study a family of problems where the goal is to make a graph Eulerian by a minimum number of deletions. We completely classify the parameterized complexity of various versions: undirected or directed graphs, vertex or edge deletions, with or without the requirement of connectivity, etc. Of particular interest is a randomized FPT algorithm for making an undirected graph Eulerian by deleting the minimum number of edges. 1
Stability type results for hereditary properties
, 2007
"... The classical Stability Theorem of Erdős and Simonovits can be stated as follows. For a monotone graph property P, let t ≥ 2 be such that t + 1 = min{χ(H) : H / ∈ P}. Then any edges from graph G ∗ ∈ P on n vertices, which was obtained by removing at most ( 1 t + o(1)) � n 2 the complete graph G = K ..."
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Cited by 3 (1 self)
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The classical Stability Theorem of Erdős and Simonovits can be stated as follows. For a monotone graph property P, let t ≥ 2 be such that t + 1 = min{χ(H) : H / ∈ P}. Then any edges from graph G ∗ ∈ P on n vertices, which was obtained by removing at most ( 1 t + o(1)) � n 2 the complete graph G = Kn, has edit distance o(n2) to Tn(t), the Turán graph on n vertices with t parts. In this paper we extend the above notion of stability to hereditary graph properties. It turns out that to do so the complete graph Kn has to be replaced by a random graph. For a hereditary graph property P, consider modifying the edges of a random graph G = G(n, 1/2) to obtain a graph G ∗ that satisfies P in (essentially) the most economical way. We obtain necessary and sufficient conditions on P which guarantee that G ∗ has a unique structure. In such cases, for a pair of integers (r, s) which depends on P, G ∗ has distance o(n2) to a graph Tn(r, s, 1 2) almost surely. Here Tn(r, s, 1 2) denotes a graph which consists of almost equal sized r + s parts, r of them induce an independent set, s induce a clique and all the bipartite graphs between parts are quasirandom (with edge density 1 2). In addition, several strengthened versions of this result are shown. 1
Hardness of edgemodification problems
"... For a graph property P consider the following computational problem. Given an input graph G, what is the minimum number of edge modifications (additions and/or deletions) that one has to apply to G in order to turn it into a graph that satisfies P? Namely, what is the edit distance ∆(G, P) of a grap ..."
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For a graph property P consider the following computational problem. Given an input graph G, what is the minimum number of edge modifications (additions and/or deletions) that one has to apply to G in order to turn it into a graph that satisfies P? Namely, what is the edit distance ∆(G, P) of a graph G from satisfying P. Clearly, the computational complexity of such a problem strongly depends on P. For over 30 years this family of computational problems has been studied in several contexts and various algorithms, as well as hardness results, were obtained for specific graph properties. Alon, Shapira and Sudakov studied in [3] the approximability of the computational problem for the family of monotone graph properties, namely properties that are closed under removal of edges and vertices. They describe an efficient algorithm that achieves an o(n2) additive approximation to ∆(G, P) for any monotone property P, where G is an nvertex input graph, and show that the problem of achieving an O(n2−ε) additive approximation is NPhard for most monotone proeprties. The methods in [3] also provide a polynomial time approximation algorithm which computes ∆(G, P)±o(n 2) for the broader family of hereditary graph properties (which are closed under removal of vertices). In this work we introduce two approaches for showing that improving upon the additive approximation achieved by this algorithm is NPhard for several subfamilies of hereditary properties. In addition, we state a conjecture on the hardness of computing the edit distance from being induced Hfree for any forbidden graph H. 1