Linear Kernels for Outbranching Problems in Sparse Digraphs

Abstract

In the k-Leaf Out-Branching and k-Internal Out-Branching problems we are given a directed graph D with a designated root r and a nonnegative integer k. The question is whether there exists an outbranching rooted at r that has at least k leaves, or at least k internal vertices, respectively. Both these problems have been studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k^2) vertices are known on general graphs. In this work we show that k-Leaf Out-Branching admits a kernel with O(k) vertices on {{mathcal {H}}}-minor-free graphs, for any fixed family of graphs {{mathcal {H}}}, whereas k-Internal Out-Branching admits a kernel with O(k) vertices on any graph class of bounded expansion.