## The involution principle

We have evaluated
the number of graphs on *v* vertices
by examining a certain action of the form _{G}Y^{X}. We shall now give
an example of the form _{H ´G}Y^{X}, i.e. a power group action.
Afterwards we shall see how these two examples can be combined in order to
prove an existence theorem for a certain class of graphs. While
doing so we shall meet an interesting and useful counting principle which
uses suitable actions of *S*_{2}, the smallest nontrivial group.

last changed: January 19, 2005