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11
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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Cited by 43 (9 self)
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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 12 (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 10 (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 ..."
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 8 (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
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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Cited by 3 (1 self)
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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
Hfree graphs of large minimum degree
 The Electronic J. Combinatorics
"... We prove the following extension of an old result of Andrásfai, Erdős and Sós. For every fixed graph H with chromatic number r +1 ≥ 3, and for every fixed ɛ>0, there are n0 = n0(H, ɛ) andρ = ρ(H)> 0, such that the following � holds. Let G be an Hfree graph on n>n0 vertices with minimum degree at le ..."
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Cited by 3 (1 self)
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We prove the following extension of an old result of Andrásfai, Erdős and Sós. For every fixed graph H with chromatic number r +1 ≥ 3, and for every fixed ɛ>0, there are n0 = n0(H, ɛ) andρ = ρ(H)> 0, such that the following � holds. Let G be an Hfree graph on n>n0 vertices with minimum degree at least 1 − 1 r−1/3 + ɛ n. Then one can delete at most n2−ρ edges to make Grcolorable. 1
collaborations and interesting discussions. Solving problems alone is boring:).
"... The first person I need to thank is my supervisor Pinar Heggernes. Without her guidance, encouragement and scolding from time to time, this work would not exist. Thank you for taking me as your student, teaching me so much and believing in me from the very start. I have never told you how much this ..."
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The first person I need to thank is my supervisor Pinar Heggernes. Without her guidance, encouragement and scolding from time to time, this work would not exist. Thank you for taking me as your student, teaching me so much and believing in me from the very start. I have never told you how much this meant to me, but I hope this thesis can make up for at least some of it. These three years gave me the opportunity to fulfill many of my dreams, and for this I will always be thankful to you. Another person to whom I owe a lot for his unconditional help, even when he hardly knew me, is Marc Bezem. Your support has been critical in many occasions, including when I had to decide whether to apply for this PhD. Thank you for convincing me to do it, or I would have regretted it forever. I would like to thank also all my coauthors Hans L. Bodlaender, Michael R.
Petr Kolman Bernard Lidick´y JeanSébastien Sereni ON FAIR EDGE DELETION PROBLEMS
"... Abstract. In edge deletion problems, we are given a graph G and a graph property π and the task is to find a subset of edges the deletion of which results in a subgraph of G satisfying the property π. Typically the objective is to minimize the total number of deleted edges while in less common fair ..."
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Abstract. In edge deletion problems, we are given a graph G and a graph property π and the task is to find a subset of edges the deletion of which results in a subgraph of G satisfying the property π. Typically the objective is to minimize the total number of deleted edges while in less common fair versions the objective is to minimize the maximum number of edges removed from a single vertex. We focus on the minimum fair odd cycle transversal (OCT) problem where the task is to make the graph bipartite; the problem is closely related to improper colorings of graphs. Though the classical version of the problem was diligently studied, the minimum fair version brings new challenges. We describe a Θ ( √ n) approximation algorithm for general graphs and an exact polynomial time algorithm for graphs of bounded treewidth. Though there are several general frameworks (e.g., MSOL) for dealing with optimization problems on graphs of bounded treewidth, the minimum fair OCT does not seem to fit into any of them. Analogous results are proved for minimum fair cut problem. 1.