Ever heard of a mathematician by name of Leonhard Euler? I recently did a video on his Seven Bridges of Köenigsburg solution and its applications to how we create poi paths...well, I've found another use for Euler's work, most notably his Polyhedron Formula. The Cliff's Notes are that Euler noticed upon studying the Platonic Solids that if you counted the number of vertices each of these polyhedra had, subtracted the total number of edges between vertices, and added back the total number of polygonal faces and the number is invariably 2. You can see an outline of the concept below:
This concept has been adapted into graph theory with the concept of a planar graph--that is an arrangement of points and line segments inlaid in a plane in such a fashion that all line segments connect to points without overlapping each other at all in the process. Needless to say, the graph definition of a face differs slightly from a polyhedra-based definition in that any area enclosed by an edge can count (so in other words, a single edge can curve back around to rejoin the vertex it came from and we'd still call the enclosed area a face).
With this in mind, here's my conjecture of the day: there is no poi path that cannot be reduced to a planar graph given the following assumptions.
I'll be working on proving this one--I hope my other math geek friends out there will do the same :) Happy computing!