We prove that given an undirected graph, one can compute, in polynomial time in <i>n</i>, a graph with at most 4<i>k</i><sup>2</sup> vertices and an integer < i>k′</i> such that it has a feedback vertex set of size at most <i>.Expand

We give a short constructive proof of a theorem of Fisher: every tournament contains a vertex whose second outneighborhood is as large as its first outneighborhood. Moreover, we exhibit two such… Expand

We prove that given an undirected graph G on n vertices and an integer k, one can compute in polynomial time in n a graph G' with at most 5k2 + k vertices, such that G has a feedback vertex set of size at most k.Expand

In this paper, we give evidence for the problems Disjoint Cycles and DisJoint Paths that they cannot be preprocessed in polynomial time such that resulting instances always have a size bounded by a Polynomial in a specified parameter (or, in short: do not have a poyleomial kernel).Expand

We consider the problem of finding a large or dense triangle-free subgraph in a given graph G. In response to a question of P. Erdős, we prove that, if the minimum degree of G is at least 9|V… Expand

We prove that for every k, there exists c k 0 such that every graph G on n vertices with no induced path P k or its complement P k contains a clique or a stable set of size n c k .Expand