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@INPROCEEDINGS{Goldreich97propertytesting,
    author = {Oded Goldreich and Dana Ron},
    title = {Property testing in bounded degree graphs},
    booktitle = {Algorithmica},
    year = {1997},
    pages = {406--415}
}

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Abstract:

We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Whereas they view graphs as represented by their adjacency matrix and measure distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded-length incidence lists and measure distance between graphs as a fraction of the maximum possible number of edges. Thus, while the previous model is most appropriate for the study of dense graphs, our model is most appropriate for the study of bounded-degree graphs. In particular, we present randomized algorithms for testing whether an unknown boundeddegree graph is connected, k-connected (for k? 1), planar, etc. Our algorithms work in time polynomial in 1=ffl, always accept the graph when it has the tested property, and reject with high probability if the graph is ffl-away from having the property. For example, the 2-Connectivity algorithm rejects (w.h.p.) any N-vertex d-degree graph for which more than ffldN edges need to be added to make the graph 2-edge-connected. In addition we prove lower bounds of \Omega\Gamma p N) on the query complexity of testing algorithms for the Bipartite and Expander properties.

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