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 size 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 soluble when one cograph is given as an induced subgraph of the other. We characterize absolute retracts of cographs.Comment: 15 page

Two remarks on retracts of graph products

AbstractLet H be a bipartie graph and let Gn be the Mycielski graph with χ(G) = n, n ⩾ 4. Then the chromatic number of the strong product of Gn by H is at most 2n − 2. We use this result to show that there exist strong products of graphs in which a projection of a retract onto a factor is not a retract of the factor. We also show that in the Cartesian product of graphs G and H, any tringles of G transfer in H, whenever G and H are connected and G is strongly-triangulated, weakly-triangulated or four-cycle free