About Assigning Roles in $P_4$-Reducible Graphs

In this work, we study the problem of assigning roles in the class of P₄-reducible graphs, a natural generalization of cographs. We show that every P₄-reducible graph with at least two vertices admits a 2-role assignment, and that every graph of this class, with the exception of P₄ itself, admits a 3-role assignment. In addition, we prove that the existence of an r-assignment in a subgraph obtained by removing an induced P₄ guarantees an (r+2)-assignment in the original graph, extending the results to higher values of r.