Contracting planar graphs to contractions of triangulations

Authors:
Marcin Kamiński;Daniël Paulusma;Dimitrios M. Thilikos
Affiliations:
Département dInformatique, Université Libre de Bruxelles, Boulevard du Triomphe CP212, B-1050 Brussels, Belgium;School of Engineering and Computing Sciences, Science Laboratories, South Road, Durham DH1 3LE, England, United Kingdom;Department of Mathematics, National and Kapodistrian University of Athens, Panepistimioupolis, GR15784 Athens, Greece
Venue:
Journal of Discrete Algorithms
Year:
2011

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Abstract

For every graph H, there exists a polynomial-time algorithm deciding if a planar input graph G can be contracted to H. However, the degree of the polynomial depends on the size of H. We identify a class of graphs C such that for every fixed H@?C, there exists a linear-time algorithm deciding whether a given planar graph G can be contracted to H. The class C is the closure of planar triangulated graphs under taking of contractions. In fact, we prove that a graph H@?C if and only if there exists a constant c"H such that if the treewidth of a graph is at least c"H, it contains H as a contraction. We also provide a characterization of C in terms of minimal forbidden contractions.