Computing role assignments of split graphs

A k-role assignment of a simple graph G is a surjective function r : V (G) -> {1,...,k} where {r(u') : u' is an element of N(u)} = {r(v') : v' is an element of N(v)} for every pair u, v is an element of V (G) with r(u) = r(v). This concept appears in the context of social networks. It is known that the problem of finding a 2-role assignment of chordal graphs can be solved in linear time and that deciding whether a graph of this class admits a k-role assignment, for any fixed k >= 3, is NP-complete. In this work, we investigate this problem on split graphs, a subclass of chordal graphs. We characterize the split graphs admitting a 3-role assignment. Such result leads to a linear time algorithm for this case. Furthermore, we prove that deciding whether a split graph has a k-role assignment is NP-complete for any fixed k >= 4. (C) 2016 Elsevier B.V. All rights reserved.