§ Discrete schild's ladder
If one is given a finite graph , which we are guaranteed came
from discretizing a grid, can we recover a global sense of orientation?
- More formally, assume the grid was of dimensions . So we have the vertex set as . We have in the edge set all elements of the form as long as respective elements and belong to .
- We lose the information about the grid as comprising of elements of the form . That is, we are no longer allowed to "look inside". All we have is a pile of vertices and edges .
- Can we somehow "re-label" the edges as "north", "south", "east", and "west" to regain a sense of orientation?
- Yes. Start with some corner. Such a corner vertex will have degree 2. Now, walk "along the edge", by going from a vertex of degree 2 to an neighbour of degree 2. If we finally reach a vertex that has unexplored neighbours of degree 3 or 4, pick the neighbour of degree 3. This will give us "some edge" of the original rectangle.
- We now arbitrary declare this edge to be the North-South edge. We now need to build the perpendicular East-West edge.