Every minor-closed property of sparse graphs is testable

Itai Benjamini, Oded Schramm, Asaf Shapira*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

45 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)2200-2218
Number of pages19
JournalAdvances in Mathematics
Volume223
Issue number6
DOIs
StatePublished - 1 Apr 2010
Externally publishedYes

Funding

FundersFunder number
National Science FoundationDMS-0901355
Directorate for Mathematical and Physical Sciences0901355

    Keywords

    • Hyper finite
    • Minor closed
    • Property testing
    • Sparse graphs

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