Contractions of planar graphs in polynomial time

Authors:
Marcin Kamiński;Daniël Paulusma;Dimitrios M. Thilikos
Affiliations:
Département d'Informatique, Université Libre de Bruxelles;Department of Computer Science, University of Durham;Department of Mathematics, National and Kapodistrian University of Athens
Venue:
ESA'10 Proceedings of the 18th annual European conference on Algorithms: Part I
Year:
2010

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Abstract

We prove that for every graph H, there exists a polynomial-time algorithm deciding if a planar graph can be contracted to H. We introduce contractions and topological minors of embedded (plane) graphs and show that a plane graph H is an embedded contraction of a plane graph G, if and only if, the dual of H is an embedded topological minor of the dual of G. We show how to reduce finding embedded topological minors in plane graphs to solving an instance of the disjoint paths problem. Finally, we extend the result to graphs embeddable in an arbitrary surface.