The isomorphism problem for graphs (GI) and the isomorphism problem for groups (GrISO) have been studied extensively by researchers. The current best algorithms for both these problems run in quasipolynomial time. In this paper, we study the isomorphism problem of graphs that are defined in terms of groups, namely power graphs, directed power graphs, and enhanced power graphs. It is not enough to check the isomorphism of the underlying groups to solve the isomorphism problem of such graphs as the power graphs (or the directed power graphs or the enhanced power graphs) of two nonisomorphic groups can be isomorphic. Nevertheless, it is interesting to ask if the underlying group structure can be exploited to design better isomorphism algorithms for these graphs. We design polynomial time algorithms for the isomorphism problems for the power graphs, the directed power graphs and the enhanced power graphs arising from finite nilpotent groups. In contrast, no polynomial time algorithm is known for the group isomorphism problem, even for nilpotent groups of class 2. We note that our algorithm does not require the underlying groups of the input graphs to be given. The isomorphism problems of power graphs and enhanced power graphs are solved by first computing the directed power graphs from the input graphs. The problem of efficiently computing the directed power graph from the power graph or the enhanced power graph is due to Cameron [IJGT'22]. Therefore, we give a solution to Cameron's question.

翻译：图的同构问题（GI）与群的同构问题（GrISO）已受到研究者的广泛研究。目前，这两种问题的最优算法均可在拟多项式时间内运行。本文研究以群定义的图的同构问题，具体包括幂图、有向幂图及增强幂图。解决此类图的同构问题不能仅通过检验底层群的同构实现，因为两个非同构群可能生成同构的幂图（或有向幂图、增强幂图）。尽管如此，探究底层群结构能否被用于设计更优的图同构算法仍然具有意义。针对有限幂零群生成的幂图、有向幂图及增强幂图，我们设计了多项式时间同构判定算法。值得注意的是，即便对于类2幂零群，其群同构问题目前仍无已知的多项式时间算法。此外，我们的算法无需输入图的底层群信息。幂图与增强幂图的同构问题通过首先从输入图中计算有向幂图来解决。从幂图或增强幂图高效计算有向幂图这一难题源于Cameron [IJGT'22]。因此，我们对Cameron的问题给出了一个解答。