A Linear Vertex Kernel for Maximum Internal Spanning Tree

Authors:
Fedor V. Fomin;Serge Gaspers;Saket Saurabh;Stéphan Thomassé
Affiliations:
Department of Informatics, University of Bergen, Norway;LIRMM, University of Montpellier 2, CNRS, France;Department of Informatics, University of Bergen, Norway;LIRMM, University of Montpellier 2, CNRS, France
Venue:
ISAAC '09 Proceedings of the 20th International Symposium on Algorithms and Computation
Year:
2009

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Abstract

We present a polynomial time algorithm that for any graph G and integer k 驴 0, either finds a spanning tree with at least k internal vertices, or outputs a new graph G R on at most 3k vertices and an integer k驴 such that G has a spanning tree with at least k internal vertices if and only if G R has a spanning tree with at least k驴 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."