A $r$-role assignment of a simple graph $G$ is an assignment of $r$ distinct roles to the vertices of $G$, such that two vertices with the same role have the same set of roles assigned to related vertices. Furthermore, a specific $r$-role assignment defines a role graph, in which the vertices are the distinct $r$ roles, and there is an edge between two roles whenever there are two related vertices in the graph $G$ that correspond to these roles. We consider complementary prisms, which are graphs formed from the disjoint union of the graph with its respective complement, adding the edges of a perfect matching between their corresponding vertices. In this work, we characterize the complementary prisms that do not admit a $3$-role assignment. We highlight that all of them are complementary prisms of disconnected bipartite graphs. Moreover, using our findings, we show that the problem of deciding whether a complementary prism has a $3$-role assignment can be solved in polynomial time.

翻译：对于一个简单图$G$，一个$r$-角色分配是指将$r$个不同角色分配给$G$的顶点，使得具有相同角色的两个顶点拥有相同的与相关顶点关联的角色集合。此外，一个具体的$r$-角色分配定义了一个角色图，其中顶点是$r$个不同角色，若图$G$中存在两个对应这些角色的相关顶点，则这两个角色之间有一条边。我们考虑互补棱柱，即由原图与其互补图的不交并，并在对应顶点之间添加完美匹配的边所构成的图。本文刻画了不允许$3$-角色分配的互补棱柱的特征。我们强调，这些互补棱柱均为不连通二分图的互补棱柱。此外，利用我们的发现，证明判断一个互补棱柱是否具有$3$-角色分配的问题可以在多项式时间内求解。