Computing some role assignments of Cartesian Product of Graphs

In social networks, a role assignment is such that individuals play the same role, if they relate in way to other playing counterpart roles. As simple graph models network rises decision problem called r -Role Assignment whether it exists an of distinct roles vertices graph. This known be NP-complete for any fixed ? 2. The Cartesian product graphs one most studied operation on and has numerous applications diverse areas, as Mathematics, Computer Science, Chemistry Biology. this paper, we determine computational complexity restricted graphs, = 2,3. fact, show always 2-role assignable, however 3-Role still class.