A linear vertex kernel for maximum internal spanning tree

Abstract

We present a polynomial time algorithm that for any graph G and integer k⩾0, either finds 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 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.”