Consider two finite subsets of a circle, $A,B \in S^1$, each with even number of elements. I want to construct two chord diagrams, one with endpoints set $A$, and one with endpoints set $B$ in such a way that no two chords of different diagrams intersect.
What are the necessary and sufficient conditions on sets $A$ and $B$ for this to be possible?
For example, it is impossible if there is an odd number of points from set $A$ between any two adjacent points of set $B$, or vice versa, but this seems like a too strong condition.
It seems like a problem that is obvious for someone knowledgeable in combinatorics and/or discrete geometry, but the full answer eludes me.