Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

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, a rational number k not smaller than 1 and a weight function w : V 7! R1 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 hG;w; ki of Weighted Max Leaf in linear time into an equivalent instance hG0;w0; k0i such that jV (G0)j  5:5k and k0  k. In the context of xed 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 = (V;E) that excludes some simple substructures always contains a spanning tree with at least jV j=5:5 leaves.

• Type: Conference
• Year: 2010
• Where: CIAC
• Authors: Bart Jansen
• Keywords: Algorithms, CIAC 2010, Positive Vertex Weights