Graphs from Seifert surfaces

Question

Given a Seifert surface if we make the disks and bands infinitely small and thin it becomes a graph where the disks are vertices and the bands are edges. Can we say that following theorem,

For every Seifert surface there is an unique graph and vice versa.

is trivial?

Answer 1

I believe that twisting a region to get more bigons doesn't change the graph but gives you a seifert surface that is not ambient-isotopic to the original surface.