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