Co-authorship for a revised but relatively new article? I am a fresh PhD. In search of postdocs and permanent positions in last 4 months, I learned about two things which evaluating/hiring committee look in their potential employees:

1) How much the candidate has grown post their PhD? If a candidate keeps publishing article with previous supervisors even after his/her PhD it may mean that they have a good working relationship, but it also means that candidate has failed to collaborate with new partners. As the present research world is considered to be highly collaborative. Also it means that even after the PhD, the candidate still needs supervision from his/her former supervisor.

2) The greater the number of coauthors are for your paper, the less weightage you get during the evaluation process.

Now main problem: One of my articles of my PhD thesis was submitted in Phys Rev Lett, and it had received one very good feedback and one very bad. That article basically consisted of an interplay of formulas and some mathematical tricks to derive a desired equation. The idea was mine and I had written the whole article by myself.

My main supervisor contribution's was that he asked me to reverse the order of two paragraphs of the final submitted version of article.  Also during most of my PhD my main-supervisor always helped and polished my work.

My co-supervisor's had initially given some ideas but I proved them to be mathematically and physically incorrect. Also during most of my PhD my co-supervisor hardly contributed anything.

Update: Now I am planning to rewrite the article and submit to the same or some another journal, but this time I will have computational fluid dynamics simulation included in the article which will prove my derivation to be correct. I assume the new article will have 50% overlap with the previous version and new 50% material will be added to it.

Question: Should I include my supervisors as co-authors, or may be just one, or may be none? I do not think I will get any input from them in this particular article.

Background: 1) Academic world acts in 3 ways I guess:

a) Do buttering of seniors, collaborate with many and cite one another, and get timely promotions by feeding on the opportunities of others. I have seen a unqualified person getting promoted from Assistant Professor to Professor in just 8 years.

b) Do not do buttering, but then suffer. I have seen a well qualified person has remained as an Assistant Professor for last 23 years.

c) Make your credentials/ profile gigantic so none could affect you significantly.

So, If I rewrite the article as a single author, it will be a great boost for my long term career.

2) But I cannot deny that a recent postdoc offer I got has some contributions from my main-supervisor since he has highly recommended me to the professor at the new institute, since both are friends. Although the professor knew my work  from before, he had attended a presentation of mine earlier, I cannot ignore the recommendation of my main supervisor. I guess that is how academia works, also it seems difficult to draw a line between buttering and seeking help. I guess when I will be looking for a permanent position after 2 years, I might need the help of my main supervisor once again?

So, how to deal with this problem?
In this particular case, I would not change the authorship for the revised version (unless there are better reasons than what you have communicated).

You must not forget to keep your personal relations oiled. If you re-submit this article as a single-author paper, i.e. remove your supervisor, your relation to them would certainly not be helped. In the future, it may be important/useful to have that relation intact and well working for ongoing/future collaborations.

Moreover, by having them as co-author they are somewhat in your debt (which cannot be a bad thing) and may return the favor by adding you to one of their papers, even if you contributed as little as they did to yours.