Testing subgraphs in directed graphs

Abstract

Let H be a fixed directed graph on h vertices, let G be a directed graph on n vertices and suppose that at least n 2 edges have to be deleted from it to make it H-free. We show that in this case G contains at least f ( , H)n h copies of H. This is proved by establishing a directed version of Szemerédi'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 P H 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 applies to two-sided error property testers as well. A special case of this result settles an open problem raised by the first author in [1]. Interestingly, it turns out that if P H has a polynomial query complexity, then there is a two-sided -tester for P H that samples only O(1/ ) vertices, whereas any one-sided tester for P H makes at least (1/ ) d/2 queries, where d is the average degree of H. We also show that the complexity of deciding if for a given directed graph H, P H has a polynomial query complexity, is N P -complete, marking an interesting distinction from the case of undirected graphs. For some special cases of directed graphs H, we describe very efficient one-sided error propertytesters for testing P H . As a consequence we conclude that when H is an undirected bipartite graph, we can give a one-sided error property tester with query complexity O((1/ ) h/2 ), improving the previously known upper bound of O((1/ ) h 2 ). The proofs combine combinatorial, graph theoretic and probabilistic arguments with results from additive number theory.

A Dirac-Type Result on Hamilton Cycles in Oriented Graphs

“…cluster V i and so (1) implies that δ(R) ≥ (δ(G)/|G| − (3ε + 2d))|R|. Altogether this shows that R is as required in the lemma.…”

“…The Diregularity lemma is a version of the Regularity lemma for digraphs due to Alon and Shapira [1]. Its proof is quite similar to the undirected version.…”

Testing satisfiability

“…This follows from the fact that any graph with at least n 2 edges contains at least Ω( t 2 n 2t ) copies of K t,t . In fact, sampling 1/ O(t) vertices suffices, by a special case of a result in [5] improving the estimate in [1]. It is also worth noting that there are simple graph properties (like the property of containing at least n 2 /4 edges) that have two-sided error testers of complexity polynomial in 1/ , but no one-sided error tester of complexity smaller that Θ(n).…”

Packing Directed Cycles Efficiently

“…The proof, which is a modified version of the proof of the standard regularity lemma given in [15], can be found in [2]. We now give the definitions necessary in order to state the directed regularity lemma.…”

“…The proof of Theorem 1.2 makes use of the recently discovered directed regularity lemma which has been proved by Alon and Shapira in [2], and which enables us to overcome several difficulties that do not occur in the undirected case.…”