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 non-isomorphic.