Fixed Parameter Complexity of the Weighted Max Leaf Problem

In this paper we consider the fixed parameter complexity of the Weighted Max Leaf problem, where we get as input an undirected connected graph G = (V,E), a set of numbers S and a weight function w : V 7→ S on the vertices, and are asked whether a spanning tree for G exists such that the combined weight of the leaves of the tree is at least k. The fixed parameter complexity of the problem strongly depends on whether we allow vertices to have a weight of 0. We show that if S = N then this problem admits a kernel with 7.5k vertices. We prove that the case S = {0} ∪ N is hard for W[1] on general graphs, but that a kernel exists with 78k vertices on planar graphs, O( √ γk+γ) vertices on graphs of oriented genus at most γ, and with (1+4δ)(1+ δ 2 )k vertices on graphs in which the degree of positive-weight vertices is bounded by δ. For the correctness proofs of the kernels we develop some extremal graph theory about spanning trees with many leaves in (bipartite) graphs without long paths of degree-2 vertices.