The Isomorphism Problem of Power Graphs and a Question of Cameron

We study the isomorphism problem of graphs that are defined in terms of groups, namely power graphs, directed power graphs, and enhanced power 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. Our algorithms do not require the underlying groups of the input graphs to be given. A crucial step in our algorithms is to reconstruct the directed power graph from the given power graph or the enhanced power graph. The problem of efficiently computing the directed power graph from a power graph or an enhanced power graph is due to Cameron [IJGT'22]. Bubboloni and Pinzauti [Arxiv'22] gave a polynomial-time algorithm to reconstruct the directed power graph from a power graph. We give an efficient algorithm to compute the directed power graph from an enhanced power graph. The tools and techniques that we design are general enough to give a different algorithm to compute the directed power graph from a power graph as well.