Here is another Petty family with Grahams (already mentioned but here in full)
1861 RG9; Piece: 2010; Folio: 100; Page: 1
Bottom of Eldon Street Darlaston
Dominick Petty head 50 ysLabourer in Colliery b Ireland
Elizabeth Petty wife 48 ys b Ireland
Job Petty son 19 ys Lock Filer (?) b England
Thomas Petty son 17 ys Key Stamper b England
Rebecca Petty daur 24 ys b Ireland
Michael Graham boarder 25 ys Coal nMiner b Ireland
Marlia Graham boarder 22 ys Coal Miner b Ireland
**Marlia looks like Martin
These Pettys are related surely - perhaps brothers? Both Jeremiah and Dominick have children called Job (Jeremiah's son b 1853) and there was a Job staying with Jeremiah in 1851.
However, I can't find these Pettys in 1851 or after 1861 which makes me wonder if
Job is the same person.

and perhaps Rebecca/Elizabeth ?