Two graphs G and H on n vertices are -far from being isomorphic if at least n 2 edges must be added or removed from E(G) in order to make G and H isomorphic. In this paper we deal with the question of how many queries are required to distinguish between the case that two graphs are isomorphic, and the case that they are -far from being isomorphic. A query is defined as probing the adjacency matrix of any one of the two graphs, i.e. asking if a pair of vertices forms an edge of the graph or not.

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... We investigate both one-sided error and two-sided error testers under two possible settings: The first setting is where both graphs need to be queried; and the second setting is where one of the graphs is fully known to the algorithm in advance. We prove that the query complexity of the best one-sided error testing algorithm is Θ(n 3/2 ) if both graphs need to be queried, and that it is Θ(n) if one of the graphs is known in advance (where the Θ notation hides polylogarithmic factors in the upper bounds). For two-sided error testers, we prove that the query complexity of the best tester is Θ( √ n) when one of the graphs is known in advance, and we show that the query complexity lies between Ω(n) and O(n 5/4 ) if both G and H need to be queried. All of our algorithms are additionally non-adaptive, while all of our lower bounds apply for adaptive testers as well as non-adaptive ones.