On retracts, absolute retracts, and foldings in cographs

A retract of a graph G is an induced subgraph H of G such that there exists a homomorphism ρ : G → H . When both G and H are cographs, we show that the problem to determine whether H is a retract of G is NP-complete; moreover, we show that this problem on cographs is fixed-parameter tractable when parameterized by the size of H . When restricted to the class of threshold graphs or to the class of trivially perfect graphs, the retract problem becomes tractable in polynomial time. The retract 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.