Suppose you have a collection of sets that is "nested":
if two sets intersect, one contains the other.
In this case, the collection can be represented by its inclusion tree: an arrow goes from a set to all of its subsets (or vice versa).
The simplest example is the collection of interval centered at 0. Its inclusion tree has a single branch.On the opposite end is the collection of disjoint sets...
This simple concept has been extensively used in analysis of gray scale images. Indeed, the level sets of the gray scale function are nested. And so are the upper and lower level sets, above. For more see Graph representation of gray scale images.
The topology graph is an alternative tool used to represent the topology of a gray scale image. It captures both upper and lower level sets in one graph. The topology graph approach is used in Pixcavator.