Bill caught me out in a mistaken use of a UK regionalism in a little story purporting to be Amercian-set: I think I did know, but had forgotten, that graft shares some meanings either side of the pond, but has some meanings that are specific to each side too. What follows are definitions courtesy of the Cambridge Advanced Learners Dictionary, accesible via OneLook:

1
graft (PIECE)
noun [C]
a piece of healthy skin or bone cut from one part of a person's body and used to repair another damaged part, or a piece cut from one living plant and fixed to another plant so that it grows there:
He has had a skin graft on his badly burned arm.

graft
verb [T]
1 to take and attach a graft:
Skin was removed from her leg and grafted on/onto her face.
2 to join or add something new:
The management tried unsuccessfully to graft new working methods onto the existing ways of doing things.



2
graft (WORK)
noun [U] UK INFORMAL
work:
I've never been afraid of hard graft.

graft
verb UK INFORMAL
to work hard:
It was very sad that after spending all those years grafting (away), he died so soon after he retired.

graft
noun [C] UK INFORMAL
a hard worker



3
graft (INFLUENCE)
noun [U] MAINLY US
the act of obtaining money or advantage through the dishonest use of political power and influence:
The whole government was riddled with graft, bribery, and corruption.





(from Cambridge Advanced Learner's Dictionary)

http://dictionary.cambridge.org/results.asp?dict=B&searchword=graft


I'll leave for others to fill in the etymological history of these derivations.