Forbidden Subgraphs of Power Graphs

The undirected power graph (or simply graph) of a group $G$, denoted by $P(G)$, is whose vertices are the elements in which two $u$ and $v$ connected an edge between if only either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number important classes, including perfect graphs, cographs, chordal split threshold can be defined structurally terms forbidden induced subgraphs. We examine each these five classes attempt to determine groups $G$ $P(G)$ lies class under consideration. give complete results case nilpotent groups, partial greater generality. In particular, always perfect; we completely (the answer same both classes). open problems.