Let's say we have a sketch of an underected graph like the one below:
And we are told to count the ways we can seat 5 people on those chairs (if the vertices are chairs). So this is a permutation problem, with the condition that we have some symmetries in the arrangement of the chairs, so that if we neglect the symmetries, we may count some cases more than once (which is not correct).
If, instead of the graph above, we had a cycle (like a ring) $C_n$, then I would say that there are $\frac12 (n-1)!$ ways to seat those people on the chairs (the vertices of $C_n$). But how can we make a permutation formula for new graph sketches like the one above?
Any idea? Are there any rules of thumb which are useful? or an algorithm?