Testing subgraphs in directed graphs

Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least n2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f( , H)nh copies of H. This is proved by establishing a directed version of Szemer?edi's regularity lemma, and implies that for every H there is a one-sided error property tester whose query complexity is bounded by a function of only for testing the property PH of being H-free. As is common with applications of the undirected regularity lemma, here too the function 1/f( , H) is an extremely fast growing function in . We therefore further prove a precise characterization of all the digraphs H, for which f( , H) has a polynomial dependency on . This implies a characterization of all the digraphs H, for which the property of being H-free has a one sided error property tester whose query complexity is polynomial in 1/ . We further show that the same characterization also a...

• Type: Conference
• Year: 2003
• Where: STOC
• Authors: Noga Alon, Asaf Shapira
• Keywords: Algorithms, Error Property Tester, One-sided Error Property, Polynomial Query Complexity, STOC 2003