Abstract
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.
Highlights
Introduction and PreliminaryRole assignments, introduced by Everett and Borgatti [1], who called them role colorings, formalize the idea, arising in the theory of social networks, that individuals of the same social role will relate in the same way to individuals playing counterpart roles.Let G be a graph with vertices representing individuals and edges representing relationships
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
Summary
Role assignments, introduced by Everett and Borgatti [1], who called them role colorings, formalize the idea, arising in the theory of social networks, that individuals of the same social role will relate in the same way to individuals playing counterpart roles. For a given graph G, we say the function r from V (G ) into the set of positive integers is a threshold role assignment (Roberts [3]) if r( x) = r( y) ⇒ d (r(N ( x)),r (N ( y))) ≤ 1. For a given graph G, we say the function r from V (G ) into the set of positive integers is a threshold close role assignment (Roberts [3]) if r ( x) = r ( y) ⇒ dh (r (N ( x)),r (N ( y))) ≤ 1. Theorem 2 A graph G with at least k vertices has a k-role assignment if and only if V (G ) can be partitioned into k nonempty sets V1,V2, ,Vk , and the following two properties are satisfied. It is easy to see that r is a (n − 1) -role assignment of graph G
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