The Mathematics Genealogy Project keeps an online database of information on as many mathematicians as they can get records for, focusing especially on advisor-student relationships. In this way, they have created a graph where the vertices are the mathematicians and the edges are the advisor-student pairings. The site tells us that the graph is non-planar, which means it's impossible to draw the graph in a plane without intersecting edges. From a graph theory perspective, the necessary and sufficient condition for a graph to be non-planar is that it contain at least one $K_5$ or $K_{3,3}$ minor, and this is what is used in the explanation on the site. But I prefer to think about this more in terms of human relationships.
A necessary (but not sufficient) condition for the graph to be non-planar is that sometimes a single student has many advisors, for if each student had exactly 0 or 1 advisors, then the graph would be finitely many disjoint trees. Here is an example where having multiple advisors causes non-planarity: two professors jointly take on three different students, then those three students become professors and jointly take on a common new student. But this example sounds unlikely.
The explanation for non-planarity given on the site is via a single example of a $K_{3,3}$ minor which contains many famous names, e.g. Gauß, Weierstraß, Klein, Hilbert, Frobenius, Kummer. If you look up the advisor-student pairings in this minor, you will find that the non-planarity is essentially caused by Frobenius having two advisors: Weierstraß and Kummer. (I wonder how different things would have been if he'd had to pick one or the other.)
My first reaction to seeing this was: are non-planar subgraphs common, or did they just pick this one so they'd have something with text-book names? And if there are other examples, do they tend to span many generations of influential mathematicians as this one does, or can we find more minimal examples like the one I constructed above?
As a more easily quantifiable measure of non-planarity, which also takes into account the condition discussed above, one could ask: how many edges need to be removed to make a planar graph, where we add the additional rule that we are not allowed to remove all of any one student's advisors? If would also be fun to have the list of name-pairs occurring as candidates for removal in this result.
To answer these questions one would need access to the database, but I wonder if this (or something similar) has already been done in attaining the information on the site.
