TY - JOUR

T1 - Every minor-closed property of sparse graphs is testable

AU - Benjamini, Itai

AU - Schramm, Oded

AU - Shapira, Asaf

N1 - Funding Information:
✩ A Preliminary version of this paper appeared in the Proc. of the 40th ACM Symposium on Theory of Computing (STOC) 2008. * Corresponding author. E-mail address: [email protected] (A. Shapira). 1 Supported in part by NSF Grant DMS-0901355.

PY - 2010/4/1

Y1 - 2010/4/1

N2 - Suppose G is a graph of bounded degree d, and one needs to remove ε{lunate}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] [20], 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.

AB - Suppose G is a graph of bounded degree d, and one needs to remove ε{lunate}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] [20], 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.

KW - Hyper finite

KW - Minor closed

KW - Property testing

KW - Sparse graphs

UR - http://www.scopus.com/inward/record.url?scp=76849098319&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2009.10.018

DO - 10.1016/j.aim.2009.10.018

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AN - SCOPUS:76849098319

SN - 0001-8708

VL - 223

SP - 2200

EP - 2218

JO - Advances in Mathematics

JF - Advances in Mathematics

IS - 6

ER -