Property Testing in Bounded Degree Graphs

We further develop the study of testing graph properties as initiated by Goldreich, Goldwasser and Ron. Loosely speaking, given an oracle access to a graph, we wish to distinguish the case when the graph has a pre‐determined property from the case when it is “far” from having this property. Whereas they view graphs as represented by their adjacency matrix and measure the distance between graphs as a fraction of all possible vertex pairs, we view graphs as represented by bounded‐length incidence lists and measure the 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 bounded‐degree graph is connected, k ‐connected (for k>1 ), cycle‐free and Eulerian. Our algorithms work in time polynomial in 1/ϵ , always accept the graph when it has the tested property, and reject with high probability if the graph is ϵ ‐far from having the property. For example, the 2‐connectivity algorithm rejects (with high probability) any N ‐vertex d ‐degree graph for which more than ϵ dN edges need to be added in order to make the graph 2‐edge‐connected.In addition we prove lower bounds of Ω(\sqrt N ) on the query complexity of testing algorithms for the bipartite and expander properties.