Retracts of strong products of graphs

Let G and H be connected graphs and let G ∗ H be the strong product of G by H. We show that every retract R of G ∗ H is of the form R = G′ ∗ H ′, where G′ is a subgraph of G and H ′ one of H. For triangle–free graphs G and H both G′ and H ′ are retracts of G and H, respectively. Furthermore, a product of finitely many finite, triangle–free graphs is retract–rigid if and only if all factors are retract–rigid and it is rigid if and only if all factors are rigid and pairwise nonisomorphic.