On Retracts, Absolute Retracts, and Folds in Cographs

Let G and H be two cographs. We show that the problem to determine whether H is a retract of G is NP-complete. We show that this problem is fixed-parameter tractable when parameterized by the order of H. When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the problem becomes tractable in polynomial time. The problem is also solvable in linear time when one cograph is given as an induced subgraph of the other. We characterize absolute retracts for the class of cographs. Foldings generalize retractions. We show that the problem to fold a trivially perfect graph onto a largest possible clique is NP-complete. For a threshold graph this folding number equals its chromatic number and achromatic number.