Computing role assignments of Cartesian product of graphs

Network science is a growing field of study using Graph Theory as a modeling tool. In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. In this sense, a role assignment permit to represent the network through a smaller graph modeling its roles. This leads to a problem called r -Role Assignment whose goal is deciding whether it exists such an assignment of r distinct roles. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is a well studied graph operation, often used for modeling interconnection networks. Formally, the Cartesian product of G and H is a graph, denoted as G□H, whose vertex set is V (G) × V (H) and two vertices (u, v) and (x, y) are adjacent precisely if u = x and vy ∈ E(H), or ux ∈ E(G) and v = y. In a previous work, we showed that Cartesian product of graphs are always 2-role assignable, however the 3-Role Assignment problem is NP-complete on this class. In this paper, we prove that r -Role Assignment restricted to Cartesian product graphs is still NP-complete, for any fixed r  ≥ 4.