I am currently a postdoc in computer science. Last year I completed my PhD. I have published one research paper during my PhD with my research supervisor. I am trying to work on that paper. I mean I have got something and I think it is going to be an incremental result of the previous research paper.
Question: Do I need to get permission from my past supervisor to publish a paper that builds on our previous co-authored paper?
If your past supervisor made a significant intellectual contribution to the new paper, then he should be a coauthor. And then, yes, you need his permission to publish the paper, because publication requires the consent of all coauthors.
If your past supervisor did not make a significant intellectual contribution to the new paper, then he should not be a coauthor. And then no, you don't need his permission to publish the paper, because he is not a coauthor.
In short: The fact that he is your former supervisor is utterly irrelevant. (This would also be true if he were your current supervisor.)
If he contributed, you have to ask him for his permission, if not, then not.
However, even if not, it would probably (depending on your relation) -- if this work is closely related to his -- be nice to tell him. I know people, who extended their PhD work after graduation and thus, unknowingly, studied the same problem the supervisor gave his new students (and thus scooped them). Your supervisor may also be able to give you valuable feedback on your draft.
