Computing role assignments of chordal graphs

In social network theory, a simple graph G is called k-role assignable if there is a surjective mapping that assigns a number from {1,…,k}, called a role, to each vertex of G such that any two vertices with the same role have the same sets of roles assigned to their neighbors. The decision problem whether such a mapping exists is called the k-Role Assignment problem. This problem is known to be NP-complete for any fixed k≥2. In this paper, we classify the computational complexity of the k-Role Assignment problem for the class of chordal graphs. We show that for this class the problem can be solved in linear time for k=2, but remains NP-complete for any k≥3. This generalizes earlier results by Sheng and answers her open problem.