I fear you are making an incorrect assumption in your question - that there is "in fact a -graphy in the general case". In fact, there is no general case that covers all words ending in -graphy in English.
As English has developed, it has borrowed many components from other languages at different times. The Greeks, due to their influential philosophy and technology, were a rich source of language roots to the rest of Europe. This process has happened continually over the centuries, and in a variety of ways, for a variety of reasons.
In English we've borrowed the Greek root γράφειν graphein "scratch, draw, write" as -graphy via Latin -graphia then French -graphie. So we have old words like cartography to modern words like photography. Wikipedia has a list of dozens of words ending in -graphy, including for example:
...but these are so distinct in English meanings that there is really no common verb describing the action of a -grapher. A calligrapher writes, a cartographer maps, a photographer photographs, a pornographer makes pornography, a radiography takes radiograms.
The common element is "writing, art, recording, listing, or a field of study". But I don't believe you'll find a single English verb that covers all that clearly.
By the way, in your answer you list construct from "draw together" as one of the meanings. This is incorrect in two ways. First, -graphy does not refer to construction except inasmuch as it refers to art or drawing. Second, you are confusing two senses of the English verb draw: one meaning is "to depict with lines"; another is "to pull". Draw together comes from the "pull" sense, not the "depict" sense.