Graphs with k-Role Assignments

For a given graph G, a k-role assignment of G is a surjective function  such that , where N(x) and N(y) are the neighborhoods of x and y, respectively. Furthermore, as we limit the number of different roles in the neighborhood of an individual, we call r a restricted size k-role assignment. When the hausdorff distance between the sets of roles assigned to their neighbors is at most 1, we call r a k-threshold close role assignment. In this paper we study the graphs that have k-role assignments, restricted size k-role assignments and k-threshold close role assignments, respectively. By the end we discuss the maximal and minimal graphs which have k-role assignments.