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a b s t r a c tA subgraph induced by k vertices is called a k-induced subgraph. We prove that determining if a digraph G contains H-free k-induced subgraphs is Ω(N 2 )-evasive. Then we construct an -tester to test this property. (An -tester for a property Π is guaranteed to distinguish, with probability at least 2/3, between the case of G satisfying Π and the case of G being -far from satisfying Π.) The query complexity of the -tester is independent of the size of the input digraph. An ( , δ)-tester for a property Π is an -tester for Π that is furthermore guaranteed to accept with probability at least 2/3 any input that is δ-close to satisfying Π. This paper presents an ( , δ)-tester for whether a digraph contains H-free k-induced subgraphs.

We study the problem of approximate non-tandem repeat (conserved regions) extraction among strings (genes). Basically, given a string S and thresholds L and D over a finite alphabet, extracting approximate repeats is to find pairs (β, β ) of substrings of S under some constraints such that β and β have edit-distance at most D and their respective lengths are at least L. Previous works mainly focus on the case that D is small, so they are not appropriate for extracting approximate repeats with relatively large D. In contrast, this paper focuses on extracting long approximate repeats with large D and it is more efficient than previous works. We also show that our algorithm is optimal in time when D is a constant.In this paper, given an input string S and thresholds L and D, we would like to extract all (D, L)-supermaximal approximate repeats (β, β ) of S. One useful application of extracting all (D, L)-supermaximal approximate repeats (β, β ) is to find all longest possible substrings β of S such that there exist some other substring β of S where β and β have edit-distance at most D and their respective lengths are at least L. This algorithm can be easily applied to the case where there are multiple input strings S1, S2, . . . , Sn if we first concatenate the input strings into one long subject string S with a special symbol " " for separation: S1 S2 . . . Sn. The running time complexity of our algorithm is O(DN 2 ) where