A Linear Vertex Kernel for Maximum Internal Spanning Tree

TL;DR

This paper introduces a polynomial-time kernelization algorithm for the Maximum Internal Spanning Tree problem, reducing it to a smaller graph while preserving the existence of a spanning tree with a specified number of internal vertices.

Contribution

It provides the first polynomial kernel of size 3k for the problem, using a novel application of hypertree and hypergraph partition connectivity results.

Findings

Established a 3k-vertex kernel for the problem

Connected hypertree existence to hypergraph partition connectivity

Designed an algorithm that either finds a spanning tree or reduces the graph size

Abstract

We present a polynomial time algorithm that for any graph G and integer k >= 0, either finds a spanning tree with at least k internal vertices, or outputs a new graph G' on at most 3k vertices and an integer k' such that G has a spanning tree with at least k internal vertices if and only if G' has a spanning tree with at least k' internal vertices. In other words, we show that the Maximum Internal Spanning Tree problem parameterized by the number of internal vertices k, has a 3k-vertex kernel. Our result is based on an innovative application of a classical min-max result about hypertrees in hypergraphs which states that "a hypergraph H contains a hypertree if and only if H is partition connected."