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.

The Erdös–Hajnal Conjecture—A Survey

“…The following is also a result from [2]; it is the directed analogue of a well-known lemma for undirected graphs [5].…”

“…In [3], the following is proved: The proof uses the directed version of Szemerédi's Regularity Lemma formulated in [2], and extensions of ideas from the proof of Theorem 3.1. Instead of starting with arbitrary sets of candidates, the way it is done in Theorem 3.1, we get a head start by using sets given by a regular partition.…”

Packing Triangles in Regular Tournaments

“…Consider, for example, the 5-vertex regular tournament obtained by the following orientation of K 5 on the vertex set {1, 2, 3, 4, 5}. Orient a Hamilton cycle (1,2,3,4,5) and another Hamilton cycle as (1,4,2,5,3). Clearly, ν 3 (T ) = 2.…”

Excluding pairs of tournaments

“…For a positive integer , we call the graph 1, a star. Let be a star with vertex set { , 1 , … , }, where is adjacent to 1 , … , . We call the center of the star, and 1 , … , the leaves of the star.…”

“…Marked vertices are the embeddings of the vertices of the stars of that were found so far. Vertex , is an embedding of the th vertex of (under ordering (1,2,3,4,5,6)) that is in the th star of and is a part of the th triple. Different vertices of the embedding are found in different sets and sets  are six-element subsets.…”