Kuratowski's Theorem tells us when we can draw a graph G in the plane without any edges crossing. It turns out G can be
so drawn exactly when it does NOT "contain" two certain graphs. What's surprising is not just that there are two such
graphs, but that the list is finite at all. Graph theorists proved many theorems in which the analogous list kept
turning out to be finite. Eventually, Robertson and Seymour of Four Color Theorem fame explained this phenomenon,
proving essentially that, "There's always such a theorem." Kuratowski's Theorem and others like it specify a property
by telling what is NOT there. Many structure theorems also characterize a property by telling what IS there. However,
proving that there is "always such a theorem" in this case appears far more difficult.


It is more difficult still to give a general procedure to determine what those structure theorems are; proving even one
specific structure theorem is normally a genuine research problem, so proving an infinite family of structure theorems
has obvious difficulties. Building on work of Robertson, Seymour, and Thomas, an algorithm was found for trees under
the topological minor relation by Nigussie to do just this. In this talk, I present an overview of the algorithm I
found last December to find infinitely many structure theorems at once for series-parallel diagrams, which generalize
trees. We give specific structure theorems also to show the algorithm is very quick by hand in practice.