A complete complexity classification of the role assignment problem

Abstract

In social network theory a society is often represented by a simple graph G, where vertices stand for individuals and edges represent relationships between those individuals. The description of the social network is tried to be simplified by assigning roles to the individuals, such that the neighborhood relation is preserved. Formally, for a fixed graph R we ask for a vertex mapping r : V G → V R , such that r ( N G ( u ) ) = N R ( r ( u ) ) for all u ∈ V G . If such a mapping exists the graph G is called R-role assignable and the corresponding decision problem is called the R-role assignment problem. Kristiansen and Telle conjectured that the R-role assignment problem is an NP -complete problem for any simple connected graph R on at least three vertices. In this paper we prove their conjecture. In addition, we determine the computational complexity of the role assignment problem for nonsimple and disconnected role graphs, as these are considered in social network theory as well.