## Every minor-closed property of sparse graphs is testable (2007)

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### BibTeX

@TECHREPORT{Benjamini07everyminor-closed,

author = {Itai Benjamini and Oded Schramm and Asaf Shapira},

title = {Every minor-closed property of sparse graphs is testable},

institution = {},

year = {2007}

}

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### Abstract

Suppose G is a graph of bounded degree d, and one needs to remove ɛn 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. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outer-planar, series-parallel, bounded genus, bounded tree-width and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. 1

### Citations

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Citation Context ...s to being a tree, was introduced by Robertson and Seymour as part of their proof of the Graph-Minor Theorem; see [7]. It also has numerous applications in the area of fixed-parameter algorithms; see =-=[17]-=- for more details. As it turns out, having bounded tree-width is also a minor-closed graph property. Another well-known minor-closed property is being Series-parallel. Series parallel graphs are graph... |

424 | Property testing and its connection to learning and approximation
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Citation Context ...s paper deals with testing properties of bounded degree graphs. Let us briefly mention some results on testing properties of dense graphs, a model that was introduced by Goldreich, Goldwasser and Ron =-=[21]-=-. In this model, a graph G is said to be ɛ-far from satisfying a property P, if one needs to add/delete at least ɛn 2 edges to/from G in order to turn it into a graph satisfying P. The tester can ask ... |

389 | A Separator Theorem for Planar Graphs
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(Show Context)
Citation Context ..., we get a tester for P, which first tests for membership in Z, and then uses the tester which can tell apart B from P \ Z. This completes the proof. Recall the Lipton-Tarjan planar separator theorem =-=[29]-=-, which says that every planar graph G has a set of vertices V0 of size O(|V (G)| 1/2 ), such that every connected component of G \ V0 has at most 2|V (G)|/3 vertices. Lipton and Tarjan used this theo... |

345 | Self-testing/correcting with applications to numerical problems
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(Show Context)
Citation Context ...ch 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 ɛ only. Blum, Luby and Rubinfeld =-=[6]-=- were the first to formulate a question of this type, and the general notion of property testing was first formulated by Rubinfeld and Sudan [37]. ∗ The Weizmann Institute and Microsoft Research † Mic... |

329 | Robust characterizations of polynomials with applications to program testing
- Rubinfeld, Sudan
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(Show Context)
Citation Context ... size is a function of ɛ only. Blum, Luby and Rubinfeld [6] were the first to formulate a question of this type, and the general notion of property testing was first formulated by Rubinfeld and Sudan =-=[37]-=-. ∗ The Weizmann Institute and Microsoft Research † Microsoft Research ‡ Microsoft Research 1sThe main focus of this paper is testing properties of graphs in the bounded degree model, which was introd... |

252 | Graph minors XIII. The disjoint paths problem - Robertson, Seymour - 1995 |

226 | Tarjan, Efficient planarity testing
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Citation Context ...estable even with o(n) queries. 1 Introduction Suppose we are given an n-vertex graph of bounded degree and are asked to decide if it is planar. This problem is well known to be solvable in time Θ(n) =-=[25]-=-. But suppose we are only asked to distinguish with probability 2/3 between the case that the input is planar from the case that an ɛfraction of its edges should be removed in order to make it planar.... |

165 |
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Citation Context ...or equivalently if P is closed under removal of edges, removal of vertices and contraction of vertices. Perhaps the most well-known result in the area of graph minors is the Kuratowski-Wagner Theorem =-=[28, 42]-=-, which states that a graph is planar if and only if it is K5-minor free and K3,3minor free. This fundamental result raised the natural question if a similar characterization, using a finite family of... |

163 |
Applications of a Planar Separator Theorem
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(Show Context)
Citation Context ...lexity of the testers as a function of ɛ. 1.3 Hyper-finiteness and testing monotone hyper-finite properties The following notion of hyper-finiteness was defined by Elek [18] (though it is implicit in =-=[30]-=-, for example). 4 The conference version of [22] announced a tester for planarity, which eventually turned out to be wrong. 3sDefinition (Hyper-Finite). A graph G = (V, E) is (δ, k)-hyper-finite if on... |

134 | The art of uninformed decisions: a primer to property testing
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- 2001
(Show Context)
Citation Context ...ent from the above discussion, our main result here is the first to show that a general (and natural) family of properties are all testable in bounded-degree graphs. For more details, see the surveys =-=[4, 12, 20, 35, 36]-=- and Section 5 where we discuss another model of testing graph properties. 1.5 Techniques and overview of the paper In the next section, we introduce a metric condition for testability. Basically, we ... |

126 |
Regular partitions of graphs
- Szemerédi
- 1978
(Show Context)
Citation Context ...t we understand testing of dense graphs more than we understand testing of bounded-degree graphs is that there are structural results “describing” dense graphs, primarily Szemerédi’s regularity lemma =-=[39]-=-, while there are no similar results for arbitrary sparse graphs. As is evident from the above discussion, our main result here is the first to show that a general (and natural) family of properties a... |

90 | A characterization of the (natural) graph properties testable with one-sided error
- Alon, Shapira
(Show Context)
Citation Context ...ether a pair of vertices, say i and j, are adjacent in the input graph G. It was shown in [21] that every partition problem like k-colorability and Max-Cut is testable in this model. Alon and Shapira =-=[2]-=- have shown that every hereditary graph property 5 is testable in dense graphs. This also gave an (essential) characterization of the graph properties that are testable with one-sided error 6 . A char... |

83 | sublinear bipartite tester for bounded degree graphs - Goldreich, A - 1999 |

78 | Property testing
- Ron
(Show Context)
Citation Context ...ent from the above discussion, our main result here is the first to show that a general (and natural) family of properties are all testable in bounded-degree graphs. For more details, see the surveys =-=[4, 12, 20, 35, 36]-=- and Section 5 where we discuss another model of testing graph properties. 1.5 Techniques and overview of the paper In the next section, we introduce a metric condition for testability. Basically, we ... |

75 | Discovering treewidth
- Bodlaender
(Show Context)
Citation Context ...e most important invariants of graphs. This notion, which measures how close a graph is to being a tree, was introduced by Robertson and Seymour as part of their proof of the Graph-Minor Theorem; see =-=[7]-=-. It also has numerous applications in the area of fixed-parameter algorithms; see [17] for more details. As it turns out, having bounded tree-width is also a minor-closed graph property. Another well... |

75 |
An extremal function for contractions of graphs
- Thomason
- 1984
(Show Context)
Citation Context ...el. In particular, minor-closed properties are trivially testable in this model with O(1/ɛ) queries and even with one-sided error. This follows from the result of Kostochka and independently Thomason =-=[26, 27, 40, 41]-=- that every finite graph with average degree at least Ω(r √ log r) contains every graph on r vertices as a minor. Therefore, every large enough finite graph with Ω(n 2 ) edges does not satisfy a minor... |

73 | Three theorems regarding testing graph properties, Random Structures and Algorithms - Goldreich, Trevisan |

68 | A combinatorial characterization of the testable graph properties: it’s all about regularity
- Alon, Fischer, et al.
(Show Context)
Citation Context ...e an (essential) characterization of the graph properties that are testable with one-sided error 6 . A characterization of the properties that are testable in dense graphs was obtained by Alon et al. =-=[1]-=-. Note that in this model (as its name suggests) we implicitly assume that the input graph is dense, because the definition of ɛ-far is relative to n 2 . Therefore, some properties are trivially testa... |

61 |
A separator theorem for nonplanar graphs
- Alon, Seymour, et al.
- 1990
(Show Context)
Citation Context ...e prove all of the stated result, with the exception of Theorem 2.2, which is proved in Section 4. In particular, by combining a theorem of Lipton and Tarjan with a result of Alon, Seymour and Thomas =-=[2]-=-, regarding separators in minor-free graphs, we get that H-minor free graphs are hyper-finite. This facilitates a proof of Theorem 1.1 from Theorem 1.2. In Section 5 we discuss several open problems a... |

59 |
Über eine eigenschaft der ebenen komplexe
- Wagner
- 1937
(Show Context)
Citation Context ...or equivalently if P is closed under removal of edges, removal of vertices and contraction of vertices. Perhaps the most well-known result in the area of graph minors is the Kuratowski-Wagner Theorem =-=[28, 42]-=-, which states that a graph is planar if and only if it is K5-minor free and K3,3minor free. This fundamental result raised the natural question if a similar characterization, using a finite family of... |

54 |
Graph minors
- Robertson, Seymour
(Show Context)
Citation Context ...fic surface. Thus the property of being embeddable in a specific surface is minor-closed. In one of the deepest results in graph theory, Robertson and Seymour proved the so called Graph-Minor Theorem =-=[34]-=-, which states that for every minor-closed graph property P, there is a finite family of graphs HP such that a graph satisfies P if and only if it is H-minor free for all H ∈ HP. Note that this in par... |

54 |
The extremal function for complete minors
- Thomason
(Show Context)
Citation Context ...el. In particular, minor-closed properties are trivially testable in this model with O(1/ɛ) queries and even with one-sided error. This follows from the result of Kostochka and independently Thomason =-=[26, 27, 40, 41]-=- that every finite graph with average degree at least Ω(r √ log r) contains every graph on r vertices as a minor. Therefore, every large enough finite graph with Ω(n 2 ) edges does not satisfy a minor... |

53 |
The minimum Hadwiger number for graphs with a given mean degree of vertices, Metody Diskret
- Kostochka
- 1982
(Show Context)
Citation Context ...el. In particular, minor-closed properties are trivially testable in this model with O(1/ɛ) queries and even with one-sided error. This follows from the result of Kostochka and independently Thomason =-=[26, 27, 40, 41]-=- that every finite graph with average degree at least Ω(r √ log r) contains every graph on r vertices as a minor. Therefore, every large enough finite graph with Ω(n 2 ) edges does not satisfy a minor... |

47 | A lower bound for the Hadwiger number of graphs by their average degree
- Kostochka
- 1984
(Show Context)
Citation Context |

45 | On testing expansion in bounded-degree graphs
- Goldreich, Ron
(Show Context)
Citation Context ... of the arguments in the present paper are motivated by some of the ideas from [10]. Another property of bounded degree graphs, which has received a lot of attention is that of being an expander, see =-=[11, 24, 27, 34]-=-. One reason for the fact that we understand testing of dense graphs more than we understand testing of bounded-degree graphs is that there are structural results “describing” dense graphs, primarily ... |

44 | Algorithmic graph minor theory: decomposition, approximation, and coloring
- Demaine, Hajiaghayi, et al.
- 2005
(Show Context)
Citation Context ... by excluded minors, and graph problems that are NP-hard in general can often be solved on excluded minor families in polynomial time, or at least be approximated better than on arbitrary graphs. See =-=[14]-=- and its references. Let us mention some well studied minor-closed graph properties. Of course, the most well known such property is Planarity. A well-studied variant of planarity is Outer-planarity, ... |

40 | Graph limits and parameter testing
- Borgs, Chayes, et al.
(Show Context)
Citation Context ... explicit upper bound on R, while the proof in [38] did not supply such a bound. We note that convergent sequences of graphs have been previously used in the study of property testing of dense graphs =-=[9]-=-. How does a tester for a monotone hyperfinite graph property P work? A first guess might be the following: as the property is monotone, we may expect to sample enough vertices such that with high pro... |

32 | A lower bound for testing 3-colorability in boundeddegree graphs
- Bogdanov, Obata, et al.
(Show Context)
Citation Context ...n bounded degree graphs some properties are testable (e.g. being triangle-free [22]), some require ˜ Θ( √ n) queries (e.g. being bipartite [22, 23]) and some require Θ(n) queries (e.g. 3-colorability =-=[8]-=-). Besides the above mentioned results, it was also shown in [22] that k-connectivity is testable in bounded degree graphs, and Czumaj and Sohler [11] have recently shown that a relaxed version of exp... |

31 |
Testing the diameter of graphs, Random structures and algorithms
- Parnas, Ron
(Show Context)
Citation Context ... every minor closed property can be tested using 2 22poly(1/ɛ) queries. 2. Another model of property testing is when the number of edges is arbitrary, and the error is relative to the number of edges =-=[32]-=-; that is, a graph with m edges is ɛ-far from P if we have to modify ɛm of its edges to get a graph satisfying P. This raises the following: Problem 5.1. What is the query complexity of testing minor-... |

29 |
Graph Theory, Third Edition
- Diestel
- 2006
(Show Context)
Citation Context ...sult deals with the testing of minor closed graph properties. Let us briefly introduce the basic notions in this area, which is too rich to thoroughly survey here. For more details, see Chapter 12 of =-=[15]-=- and the recent article of Lovász [31] on the subject. A graph H is said to a minor of a graph G, if H can be obtained from G using a sequence of vertex removals, edge removals and edge contractions 3... |

24 | Sublinear time algorithms
- Rubinfeld
- 2006
(Show Context)
Citation Context ...ent from the above discussion, our main result here is the first to show that a general (and natural) family of properties are all testable in bounded-degree graphs. For more details, see the surveys =-=[4, 12, 20, 35, 36]-=- and Section 5 where we discuss another model of testing graph properties. 1.5 Techniques and overview of the paper In the next section, we introduce a metric condition for testability. Basically, we ... |

20 | Testing hereditary properties of non-expanding bounded-degree graphs, submitted (full version of [13
- Czumaj, Shapira, et al.
(Show Context)
Citation Context ...tivity is testable in bounded degree graphs, and Czumaj and Sohler [11] have recently shown that a relaxed version of expansion is testable with ˜ Θ( √ n) queries. Finally, Czumaj, Sohler and Shapira =-=[10]-=- have recently shown that every hereditary property is testable if the input graph is guaranteed to be non-expanding. Some of the arguments in the present paper are motivated by some of the ideas from... |

15 | Testing expansion in bounded-degree graphs
- Czumaj, Sohler
(Show Context)
Citation Context ...and some require Θ(n) queries (e.g. 3-colorability [8]). Besides the above mentioned results, it was also shown in [22] that k-connectivity is testable in bounded degree graphs, and Czumaj and Sohler =-=[11]-=- have recently shown that a relaxed version of expansion is testable with ˜ Θ( √ n) queries. Finally, Czumaj, Sohler and Shapira [10] have recently shown that every hereditary property is testable if ... |

12 | Sublinear-time algorithms
- Czumaj, Sohler
(Show Context)
Citation Context |

9 | A Regularity lemma for bounded degree graphs and its applications: parameter testing and infinite volume limits
- Elek
- 711
(Show Context)
Citation Context ...s at most some fixed function g(r) satisfying g(r) = 2 o(r) as r → ∞. Theorem 1.2 can be used to show that this property is testable (but this does not seem to follow directly from Theorem 1.1). Elek =-=[19]-=- studied property testing within this class of graphs. 1.4 Comparison to previous results Our main result in this paper deals with testing properties of bounded degree graphs. Let us briefly mention s... |

9 | Homomorphisms in graph property testing - a survey
- Shapira, Alon
- 2005
(Show Context)
Citation Context ...ent from the above discussion, our main result here is the first to show that a general (and natural) family of properties are all testable in bounded-degree graphs. For more details, see the surveys =-=[3, 12, 20, 35, 36]-=- and Section 5 where we discuss another model of testing graph properties. 1.5 Techniques and overview of the paper In the next section, we introduce a metric condition for testability. Basically, we ... |

8 | A separation theorem in property testing - Alon, Shapira |

8 | Testing the expansion of a graph
- Nachmias, Shapira
- 2007
(Show Context)
Citation Context ... of the arguments in the present paper are motivated by some of the ideas from [10]. Another property of bounded degree graphs, which has received a lot of attention is that of being an expander, see =-=[11, 24, 27, 34]-=-. One reason for the fact that we understand testing of dense graphs more than we understand testing of bounded-degree graphs is that there are structural results “describing” dense graphs, primarily ... |

6 | On testable properties in bounded degree graphs - Czumaj, Sohler |

6 | Algorithmic graph minor theory: Improved grid minor bounds and wagnerâ Ă´ Zs contraction
- Demaine, Hajiaghayi, et al.
(Show Context)
Citation Context ... by excluded minors, and graph problems that are NP-hard in general can often be solved on excluded minor families in polynomial time, or at least be approximated better than on arbitrary graphs. See =-=[14]-=- and its references. Let us mention some well studied minor-closed graph properties. Of course, the most well known such property is Planarity. A well-studied variant of planarity is Outer-planarity, ... |

6 |
Property Testing in Bounded-Degree Graphs
- Goldreich, Ron
- 1997
(Show Context)
Citation Context ... property testing was first formulated by Rubinfeld and Sudan [37]. The main focus of this paper is testing properties of graphs in the bounded degree model, which was introduced by Goldreich and Ron =-=[22]-=-. In this model, we fix a degree bound d and represent graphs using adjacency lists. More precisely, we assume that a graph G is represented as a function fG : [n] × [d] ↦→ [n] ∪ {∗}, where given a ve... |

5 |
Homomorphisms in graph property testing
- Alon, Shapira
- 2006
(Show Context)
Citation Context |

5 | The combinatorial cost
- Elek
(Show Context)
Citation Context ... upper bounds on the query complexity of the testers as a function of ɛ. 1.3 Hyper-finiteness and testing monotone hyper-finite properties The following notion of hyper-finiteness was defined by Elek =-=[18]-=- (though it is implicit in [30], for example). 4 The conference version of [22] announced a tester for planarity, which eventually turned out to be wrong. 3sDefinition (Hyper-Finite). A graph G = (V, ... |

4 |
Property Testing in Bounded-Degree Graphs, Algorithmica 32
- Goldreich, Ron
- 2002
(Show Context)
Citation Context ... and Microsoft Research † Microsoft Research ‡ Microsoft Research 1sThe main focus of this paper is testing properties of graphs in the bounded degree model, which was introduced by Goldreich and Ron =-=[22]-=-. In this model, we fix a degree bound d and represent graphs using adjacency lists. More precisely, we assume that a graph G is represented as a function fG : [n] × [d] ↦→ [n] ∪ {∗}, where given a ve... |

3 |
Testing expansion in bounded-degree graphs
- Kale, Seshadhri
- 2007
(Show Context)
Citation Context ... of the arguments in the present paper are motivated by some of the ideas from [10]. Another property of bounded degree graphs, which has received a lot of attention is that of being an expander, see =-=[11, 24, 27, 34]-=-. One reason for the fact that we understand testing of dense graphs more than we understand testing of bounded-degree graphs is that there are structural results “describing” dense graphs, primarily ... |

3 |
The combinatorial cost, lEnseignement
- Elek
- 2007
(Show Context)
Citation Context ... upper bounds on the query complexity of the testers as a function of ɛ. 1.3 Hyper-finiteness and testing monotone hyper-finite properties The following notion of hyper-finiteness was defined by Elek =-=[17]-=- (though it is implicit in [32], for example). Definition (Hyper-Finite). A graph G = (V, E) is (δ, k)-hyper-finite if one can remove δ|V | edges from G and obtain a graph with connected components of... |

3 |
Parameter testing with bounded degree graphs of subexponential growth
- Elek
- 2010
(Show Context)
Citation Context ...h graphs are said to have sub-exponential growth. Then, Theorem 1.2 can be used to show that this property is testable. This does not seem to follow directly from Theorem 1.1. 1.4 Recent results Elek =-=[18]-=- has studied the problem of testing properties of graphs of sub-exponential growth (defined at the end of the previous subsection). Besides being able to handle some hereditary properties like the one... |

2 | The Complexity of Learning Minor-Closed Graph Classes - Domingo, Shawe-Taylor - 1995 |

2 |
Graph sequences with hyperfinite limits are hyperfinite
- Schramm
- 711
(Show Context)
Citation Context ...phs are far away in some pseudometric ρR from graphs that are not (ɛ ′ , k)-hyper-finite, where ɛ ′ depends on ɛ and tends to zero as ɛ ↘ 0. This result can be deduced from a recent result of Schramm =-=[38]-=-, concerning properties of convergent sequences of bounded degree graphs. However, we provide an alternative self-contained proof in Section 4. In contrast with [38], our proof is finitary and gives a... |

2 |
Property Testing
- Goldreich, Ron
(Show Context)
Citation Context ...s initiated by Goldreich Goldwasser and Ron [21]. The main focus of this paper is testing properties of graphs in the bounded degree model, which was first introduced and studied by Goldreich and Ron =-=[22]-=-. In this model, we fix a degree bound d and represent graphs using adjacency lists. More precisely, we assume that a graph G is represented as a function fG : [n] × [d] ↦→ [n] ∪ {∗}, where given a ve... |