We present a polynomial time algorithm that for any graph G and integer k ≥ 0, either nds a spanning tree with at least k internal vertices, or outputs a new graph GR on at most 3k vertices and an integer k such that G has a spanning tree with at least k internal vertices if and only if GR has a spanning tree with at least k internal vertices. In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel. Our

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... sult is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that a hypergraph H contains a hypertree if and only if H is partition connected.