Edge contractions in subclasses of chordal graphs

  • Authors:

  • Rémy Belmonte;Pinar Heggernes;Pim van 't Hof

  • Affiliations:

  • Department of Informatics, University of Bergen, Bergen, Norway;Department of Informatics, University of Bergen, Bergen, Norway;Department of Informatics, University of Bergen, Bergen, Norway

  • Venue:
  • TAMC'11 Proceedings of the 8th annual conference on Theory and applications of models of computation
  • Year:
  • 2011

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Abstract

Modifying a given graph to obtain another graph is a wellstudied problem with applications in many fields. Given two input graphs G and H, the Contractibility problem is to decide whether H can be obtained from G by a sequence of edge contractions. This problem is known to be NP-complete already when both input graphs are trees of bounded diameter. We prove that CONTRACTIBILITY can be solved in polynomial time when G is a trivially perfect graph and H is a threshold graph, thereby giving the first classes of graphs of unbounded treewidth and unbounded degree on which the problem can be solved in polynomial time. We show that this polynomial-time result is in a sense tight, by proving that Contractibility is NP-complete when G and H are both trivially perfect graphs, and when G is a split graph and H is a threshold graph. If the graph H is fixed and only G is given as input, then the problem is called H-CONTRACTIBILITY. This problem is known to be NP-complete on general graphs already when H is a path on four vertices. We show that, for any fixed graph H, the H-CONTRACTIBILITY problem can be solved in polynomial time if the input graph G is a split graph.