Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

Abstract

In this paper we consider a natural generalization of the well-known Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G = (V,E), a rational number k ≥ 1 and a weight function $w: V \longmapsto Q_{\geq 1}$ on the vertices, and are asked whether a spanning tree T for G exists such that the combined weight of the leaves of T is at least k. We show that it is possible to transform an instance 〈G,w, k 〉 of Weighted Max Leaf in linear time into an equivalent instance 〈G′,w′, k′ 〉 such that |V′| ≤ 5.5k′ and k′ ≤ k. In the context of fixed parameter complexity this means that Weighted Max Leaf admits a kernel with 5.5k vertices. The analysis of the kernel size is based on a new extremal result which shows that every graph G that excludes some simple substructures always contains a spanning tree with at least |V|/5.5 leaves.