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The parameterized Multicut problem consists in deciding, given a graph, a set of requests (i.e. pairs of vertices) and an integer k, whether there exists a set of k edges which disconnects the two endpoints of each request. Determining whether Multicut is Fixed-Parameter Tractable with respect to k is one of the most important open question in parameterized complexity [5]. We show that Multicut reduces to instances of treewidth bounded in k. To that aim, we establish new reduction rules that apply to arbitrary instances of Multicut. Based on graph separability properties, these rules identify an irrelevant request that can be safely removed. As a main consequence, these rules imply that the degree of the request graph of any instance is bounded by a function of k. We prove that when the input graph has a large clique minor or a large grid minor, then we can remove an irrelevant request or contract an edge

Year: 2010

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