For two Rubik cubes to be identical, what is the minimum requirements? for example are 2 faces and their position sufficient? e.g. 2 side by side face, or two opposing faces of two cubes being same would be sufficient for two cubes to be identical?
What about a square from not the middle of all faces being specified with their relative positions, for example would specifying bottom left hand side of each face be sufficient? e.g. on Top Red , on Bottom Blue, on left face Green, ... etc, would that be easy to see whether two cubes are identical.
Is there a name for minimum set of requirements that makes two structures isomorphic/identical to each other?
Just considering cubie permutations (ignoring their individual rotations), the corner and edge cubies together have exactly $\frac{8!12!}2$ valid configurations so you'd have to identify 7 out of the 8 corners, 10/12 edges. and two adjacent centers (to check the total orientation) to uniquely identify the permutation of all the cubies. Then you'd have to examine the rotations of 7 out of the 8 corners and 11 out of the 12 edges to uniquely identify all the cubie rotations.