Median and quasi-median direct products of graphs

Median graphs are characterized among direct products of graphs on at least three vertices. Beside some trivial cases, it is shown that one component of G × P3 is median if and only if G is a tree in that the distance between any two vertices of degree at least 3 is even. In addition, some partial results considering median graphs of the form G ×K2 are proved, and it is shown that the only nonbipartite quasimedian direct product is K3 ×K3.