Testing the diameter of graphs
Random Structures & Algorithms
Testing subgraphs in large graphs
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
A characterization of easily testable induced subgraphs
SODA '04 Proceedings of the fifteenth annual ACM-SIAM symposium on Discrete algorithms
Tight Bounds for Testing Bipartiteness in General Graphs
SIAM Journal on Computing
Testing subgraphs in directed graphs
Journal of Computer and System Sciences - Special issue: STOC 2003
Repeated communication and Ramsey graphs
IEEE Transactions on Information Theory
On testable properties in bounded degree graphs
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Comparing the strength of query types in property testing: the case of testing k-colorability
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
APPROX '07/RANDOM '07 Proceedings of the 10th International Workshop on Approximation and the 11th International Workshop on Randomization, and Combinatorial Optimization. Algorithms and Techniques
Property Testing: A Learning Theory Perspective
Foundations and Trends® in Machine Learning
Algorithmic and Analysis Techniques in Property Testing
Foundations and Trends® in Theoretical Computer Science
Introduction to testing graph properties
Property testing
Introduction to testing graph properties
Property testing
Introduction to testing graph properties
Studies in complexity and cryptography
Distance approximation in bounded-degree and general sparse graphs
APPROX'06/RANDOM'06 Proceedings of the 9th international conference on Approximation Algorithms for Combinatorial Optimization Problems, and 10th international conference on Randomization and Computation
Testing the (s,t)-disconnectivity of graphs and digraphs
Theoretical Computer Science
Principles of network computing
TAMC'12 Proceedings of the 9th Annual international conference on Theory and Applications of Models of Computation
A space efficient streaming algorithm for triangle counting using the birthday paradox
Proceedings of the 19th ACM SIGKDD international conference on Knowledge discovery and data mining
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In this paper we consider the problem of testing whether a graph is triangle-free, and more generally, whether it is H-free, for a fixed subgraph H. The algorithm should accept graphs that are triangle-free and reject graphs that are far from being triangle-free in the sense that a constant fraction of the edges should be removed in order to obtain a triangle-free graph. The algorithm is allowed a small probability of error.This problem has been studied quite extensively in the past, but the focus was on dense graphs, that is, when d = Θ(n), where d is the average degree in the graph and n is the number of vertices. Here we study the complexity of the problem in general graphs, that is, for varying d.Our main finding is a lower bound of Ω(n1/3) on the necessary number of queries that holds for every d n1-ν(n), where ν(n) = o(1). Since when d = Θ(n) the number of queries sufficient for testing has been known to be independent of n, we observe an abrupt, threshold-like behavior of the complexity of testing around n. This lower bound holds for testing H-freeness of every non-bipartite subgraph H.Additionally we provide sub-linear upper bounds for testing triangle-freeness that are at most quadratic in the stated lower bounds, and we describe a transformation from certain one-sided error lower bounds for testing subgraph-freeness to two-sided error lower bounds.Finally, in the course of our analysis we show that dense random Cayley graphs behave like quasi-random graphs in the sense that relatively large subsets of vertices have the "correct" edge density. The result for subsets of this size cannot be obtained from the known spectral techniques that only supply such estimates for much larger subsets.