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A 4k2 kernel for feedback vertex set
TLDR
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
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Median orders of tournaments: A tool for the second neighborhood problem and Sumner's conjecture
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 suchExpand
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Multicut is FPT
TLDR
We show that there exists an O(k)nc algorithm which decides if there exists a multicut of size at most k. Expand
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Partitioning a graph into a cycle and an anticycle, a proof of Lehel's conjecture
TLDR
We prove that every graph G has a vertex partition into a cycle and an anticycle (a cycle in the complement of G). Expand
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A quadratic kernel for feedback vertex set
TLDR
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
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Kernel Bounds for Disjoint Cycles and Disjoint Paths
TLDR
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
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Oriented Hamiltonian Paths in Tournaments: A Proof of Rosenfeld's Conjecture
TLDR
We prove that with three exceptions, every tournament of order n contains each oriented path ofOrder n. Expand
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Density Conditions For Triangles In Multipartite Graphs
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|VExpand
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The Erdős-Hajnal conjecture for paths and antipaths
TLDR
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
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Kernel bounds for disjoint cycles and disjoint paths
TLDR
In this paper, we show that the problems Disjoint Cycles and DisJoint Paths do not have polynomial kernels, unless NP@?coNP/poly. Expand
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