  From Smyth's classification, modular compactifications of pointed smooth
rational curves are indexed by combinatorial data, so-called extremal
assignments. We explore their combinatorial structures and show that any
extremal assignment is a finite union of atomic extremal assignments. We
discuss a connection with the birational geometry of the moduli space of stable
pointed curves. As applications, we study three special classes of extremal
assignments: smooth, toric, and invariant with respect to the symmetric group
action. We identify them with three combinatorial objects: simple intersecting
families, complete multipartite graphs, and special families of integer
partitions, respectively.
