The intersection graph of an orientable generic surface

I answer an open question left by Gui-Song Li in "On self-intersections of immersed surfaces" (AMS Proceedings, Volume 126, 1998, pp.3721-3726.) The intersection graph $M(i)$ of a generic surface $i:F \to S^3$ is the set of values which are either singularities or intersections. It is a multigraph whose edges are transverse intersections of two surfaces and whose vertices are triple intersections and cross-caps. $M(i)$ has an additional structure which Li called "a daisy graph." If F is oriented then the orientation further refines $M(i)$'s structure into what Li called an "arrowed daisy graph." Li left the open question "which arrowed daisy graphs can be realized as the intersection graph of an oriented generic surface?" The main theorem of this article will answer this. I will also provide some generalizations and extensions to this theorem in sections 4 and 5.