On Contracting Graphs to Fixed Pattern Graphs

Authors:
Pim Hof;Marcin Kamiński;Daniël Paulusma;Stefan Szeider;Dimitrios M. Thilikos
Affiliations:
School of Engineering and Computing Sciences, University of Durham, Science Laboratories, Durham, England DH1 3LE;Computer Science Department, Université Libre de Bruxelles, Brussels, Belgium B-1050;School of Engineering and Computing Sciences, University of Durham, Science Laboratories, Durham, England DH1 3LE;School of Engineering and Computing Sciences, University of Durham, Science Laboratories, Durham, England DH1 3LE;Department of Mathematics, National and Kapodistrian University of Athens, Athens, Greece GR15784
Venue:
SOFSEM '10 Proceedings of the 36th Conference on Current Trends in Theory and Practice of Computer Science
Year:
2009

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Abstract

For a fixed graph H, the H-Contractibility problem asks if a graph is H-contractible, i.e., can be transformed into H via a series of edge contractions. The computational complexity classification of this problem is still open. So far, H has a dominating vertex in all cases known to be polynomially solvable, whereas H does not have such a vertex in all cases known to be NP-complete. Here, we present a class of graphs H with a dominating vertex for which H-Contractibility is NP-complete. We also present a new class of graphs H for which H-Contractibility is polynomially solvable. Furthermore, we study the (H,v)-Contractibility problem, where v is a vertex of H. The input of this problem is a graph G and an integer k. The question is whether G is H-contractible such that the "bag" of G corresponding to v contains at least k vertices. We show that this problem is NP-complete whenever H is connected and v is not a dominating vertex of H.