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, Postboks 7803, 5020 Bergen, Norway;Institute of Information Systems (184/3), Vienna University of Technology, Favoritenstraíe 9-11, A-1040 Vienna, Austria;Institute of Mathematical Sciences, CIT Campus, Taramani, 600 113 Chennai, India;LIP, ENS Lyon, 46 allée dItalie, 69364 Lyon Cedex 07, France
Venue:
Journal of Computer and System Sciences
Year:
2013

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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"R 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"R 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.''