Linear kernels for outbranching problems in sparse digraphs

TL;DR

This paper demonstrates that specific outbranching problems in sparse directed graphs admit linear kernels, improving kernel size bounds for certain graph classes in parameterized complexity.

Contribution

It establishes linear kernels for the $k$-Leaf and $k$-Internal Out-Branching problems on $ ext{H}$-minor-free graphs and graphs of bounded expansion, respectively.

Findings

Linear kernels with $O(k)$ vertices for $k$-Leaf Out-Branching on $ ext{H}$-minor-free graphs.

Linear kernels with $O(k)$ vertices for $k$-Internal Out-Branching on graphs of bounded expansion.

Improved kernelization bounds 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 to determine the existence of an outbranching rooted at $r$ that has at least $k$ leaves, or at least $k$ internal vertices, respectively. Both these problems were intensively 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.