Edge contractions in subclasses of chordal graphsRémy Belmonte, Pinar Heggernes and Pim van 't Hof
A preliminary version of this paper appeared in the proceedings of TAMC 2011, the 8th Annual Conference on Theory and Applications of Models of Computation (to be held on May 23-25, 2010, Tokyo, Japan), Lecture Notes in Computer Science, vol. 6648, pp. 528-539, 2011.
This paper was also presented at the DIMAP Workshop on Combinatorics and Graph Theory (held on April 4-7, 2011 at the University of Warwick, UK).
Modifying a given graph to obtain another graph is a well-studied 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.