I am thinking about graph automorphisms, which contain at most one transposition, let's call them A1. I.e. two vertices can be swapped and we will still have the same graph.
Let $R(u,v) \iff $ transposition of u,v is an automorphism. The relation R is equivalence. It partitions the vertices into equivalency classes. Composition of automorphisms from A1 (closure) generates all permutations, that mix elements only within these equivalency classes.
Such automorphisms can be represented as a table, which attaches the equivalency class to each vertex ("in linear memory"). E.g. I can give each vertex a color of its class, and then I know, that I can mix the vertices with the same color any way I want, and it will still be an automorphism.
Now I am thinking about automorphisms consisting of at most two disjoint transpositions, let's call them A2. Two pairs should be swapped to get the same graph.
Is there any analogy of R for A2, or even an equivalence relation? Would it be possible to represent such automoprhisms efficiently (maybe within quadratic memory)?
Do these types of automorphisms already have names in Math?
P.S. What is my goal: I am making a computer program. The input will be an oriented graph and k-tuples of vertices of the graph, like $a_1=(1,3,5,2,2)$, $a_2=(4,5,3,1,2)$, ... . Tuple $a_i $ is equivalent to $a_j$ when there is a graph automorphism P, such that $\forall h: P(a_i[h]) = a_j[h]$. I need to remove tuples, that are equivalent to other tuples. The ultimate goal would be to keep just one representative from each equivalence class.
In other words, I need to filter out my list of tuples as much as possible. I can already detect closure(A1) automorphisms efficiently, and I would like to extend it with more kinds of automorphisms.