Should the supervisor be involved in writing parts of the publication with their students? correct parts the student writes or shoould the supervisor just submit the paper after the student has complete with writing it? which is more common in the EU CS labs?
Well, it really depends on circumstances - the student, how far along in their studies they are, and on the philosophy and the workload of the professor.
Based on what I do with my own students and see other professors do, I would say the most common trajectory in applied CS labs in the parts of Europe I worked in (Austria, Switzerland, Sweden) is roughly as follows ("supervisors" may include postdocs or other more senior lab members that mentor the student):
- For the first one or two papers, a lot of very direct support is provided. Supervisors sit with the student frequently, give rapid feedback on short text passages (subsections or even paragraphs), and write or re-write good chunks of the paper themselves.
- For later papers, the supervisors (intentionally) get more hands-off. Drafts are of course still read, but on a more coarse-grained level (say, section by section after the student has written what they believe to be a mature draft). Most, though typically not all, text in the final manuscript is predominantly written by the student.
- Given that the goal of a PhD education is to train independent researchers, the hope is that by the end a student can write a more or less submittable paper without much input by the supervisors at all. Supervisors may still help polish the final papers a bit, or give feedback on a more strategic level - or they might drop out entirely and ask the by-then mature student to submit the paper alone or with other collaborators.
I would say "submitting the paper" is the student's job in all phases. This is a fairly boring and, sometimes, tedious task, and I see no particular reason why it's a good use of valuable supervisor time to do that. I do sometimes sit with the student and do it together with them if they are taunted by a particularly obnoxious submission system.
Of course there is substantial deviations from this, both between labs and between individual students. To be frank, many students in practice never learn to write well enough to ever fly solo in their PhD education, and they never leave the second phase. Some professors will start directly in the second phase, expecting somewhat mature writing from the get-go. Some supervisors have no problem with writing or revising some sections of the paper for a student, others categorically do not do that and only comment on the student's writing.
It of course depends. Sometimes, the role of the supervisor is limited to finding the funding, proposing the work, supervising the work, checking the article to be submitted and correct or ask for corrections, and approving the publication. Strictly speaking, just funding and proposing the work is not enough for authorship, but this rule is probably violated a lot. There are also many cases where the supervisor is more involved.
For first time authors, it is very common to underestimate the contribution of a supervisor.
Since you are in CS, combined authorship is much more common than in pure Mathematics, where the rule seems to be more that everyone needs to contribute at least a result (Lemma, Theorem) and Ph.D. supervisors are usually not a co-author.
I think the answer to the question is that the supervisor should be involved in all parts of study design, study execution, and the writing process. That is, after all, what is commonly considered necessary to be a coauthor.
That is specifically true for papers written with junior authors who surely all can use the mentoring that comes with involvement in writing papers.
Of course, what I think you are really asking is "what do I do if the supervisor is not involved". But that's a different question, and one that has many previous answers on this site.