Let's say we have an overhead view of a pile of $N$ (straight) sticks of nonzero thickness, like so. Assume that each of the $I$ intersections involves just two sticks.
Now the picture as given has no depth information, and at each intersection we have two possibilities for which stick lies on top: stick A or stick B. For the diagram overall we have $2^I$ ways we can draw these over/under assignments. However, we find that not all such assignments are valid (i.e., physically realizable). Here are some invalid assignments for $N=4$ and $N=6$; we can intuitively see that these situations are not possible with straight sticks.
The question is: Given a stick pile and a set of over/under assignments, how can we determine whether that assignment is valid?
This is a simple enough problem that I'm sure someone must have looked at it, but I don't know where to begin!
EDITS FOR CLARITY:
- as shown in the $N=6$ drawing, the sticks are of finite length, and not all stick pairs intersect when viewed from above
- examples of a valid assignment would be either of the $N=4$ or $N=6$ drawings, but with any single intersection flipped (i.e., the "under" stick goes on top)
- an assignment provides an orientation to all of the $I$ intersections in the diagram, not just some; intersections cannot remain "indeterminate"
- the precise physical locations of the sticks are (I believe) unimportant, and sticks can be moved as long as the intersections don't appear, disappear, or move over one another (this invariance is actually a guess on my part, I haven't proved it)