Computing role assignments of Cartesian product of graphs

D. Castonguay, Elisângela Silva Dias, Fernanda Neiva Mesquita, J. R. Nascimento
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Abstract

Network science is a growing field of study using Graph Theory as a modeling tool. In social networks, a role assignment is such that individuals play the same role, if they relate in the same way to other individuals playing counterpart roles. In this sense, a role assignment permit to represent the network through a smaller graph modeling its roles. This leads to a problem called r -Role Assignment whose goal is deciding whether it exists such an assignment of r distinct roles. This problem is known to be NP-complete for any fixed r ≥ 2. The Cartesian product of graphs is a well studied graph operation, often used for modeling interconnection networks. Formally, the Cartesian product of G and H is a graph, denoted as G□H, whose vertex set is V (G) × V (H) and two vertices (u, v) and (x, y) are adjacent precisely if u = x and vy ∈ E(H), or ux ∈ E(G) and v = y. In a previous work, we showed that Cartesian product of graphs are always 2-role assignable, however the 3-Role Assignment problem is NP-complete on this class. In this paper, we prove that r -Role Assignment restricted to Cartesian product graphs is still NP-complete, for any fixed r  ≥ 4.
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图的笛卡尔积的角色分配计算
网络科学是一门以图论为建模工具的新兴研究领域。在社会网络中,角色分配是这样的,如果个人以同样的方式与扮演对等角色的其他个人联系在一起,他们就扮演同样的角色。从这个意义上说,角色分配允许通过对其角色建模的更小的图来表示网络。这就导致了一个叫做r角色分配的问题,其目标是确定是否存在r不同角色的分配。对于任意固定的r≥2,这个问题已知是np完全的。图的笛卡尔积是一种研究得很好的图运算,经常用于互连网络的建模。形式上,G与H的笛卡尔积是一个图,记作G□H,其顶点集为V (G) × V (H),当u = x且vy∈E(H),或ux∈E(G)且V = y时,两个顶点(u, V)和(x, y)精确相邻。在之前的工作中,我们证明了图的笛卡尔积总是2角色可分配的,但3角色分配问题在这类上是np完全的。本文证明了对于任意固定的r≥4,限制于笛卡尔积图的r -角色分配仍然是np完全的。
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