Jerrum-reduced generating set

Definition
Let $$G$$ be a group acting faithfully on a set $$S$$ of size $$n$$ (equivalently, $$G$$ is a group embedded into the symmetric group on $$n$$ elements). A Jerrum-reduced generating set for $$G$$ is a generating set for $$G$$ such that the following graph associated with the generating set is acyclic:


 * The vertices of the graph are elements of the set $$S$$ on which $$G$$ acts. Assume $$S = \{ 1 ,2, 3, \ldots, n \}$$
 * For each element $$g$$ of the generating set, there is one edge from the smallest element $$s \in S$$ such that $$g.s \ne s$$, to $$g.s$$.

Note that the notion of Jerrum-reduced depends on a total ordering of the set on which the group acts. In particular, it is possible for a generating set to be Jerrum-reduced for one choice of ordering but to not be Jerrum-reduced for some other choice of ordering.

Jerrum's filter is an online algorithm which gives a process of starting from any generating set and Jerrum-reducing it. Moreover, for a generating set that is already Jerrum-reduced, the algorithm proceeds virtually instantaneously.

Stronger properties

 * Weaker than::Permutation-invariantly Jerrum-reduced generating set: This is a generating set that is Jerrum-reduced for any choice of ordering of the underlying set it is acting on.