A Linear Vertex Kernel for Maximum Internal Spanning Tree

We present an algorithm that for any graph G and integer k ≥ 0 in time polynomial in the size of G 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 parameterized Maximum Internal Spanning Tree problem with parameter k being the number of internal vertices, has a 3k-vertex kernel. Our result 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.