Is there a number of responsibilities that fall under UX (for example, graphic design, usability etc), or is UX a sub responsibility of design?
The short answer: Design can be one of the responsibilities that fall under the larger category of UX.
According the A Project Guide to UX Design, a UX designer generally fulfills the rolls of three other categories of job, which may have separate people to perform them in extraordinarily large companies:
According to that same book, the UX designer may also interact with (or in some companies act as) the following six roles.
So a UX designer won't necessarily have any experience in graphic design, but it's common for someone working in UX to have at least some graphic design knowledge.
I think the job of taking care of User Experience utilizes graphic design, programming, and business logic to create an over-all look feel that is consistent and usable (obviously).
Perhaps it is more of a position that guides the direction of an entire project's look and feel (Rather than actual technical work). Yet it certainly could entail such things if you are on a small team.
The specific role of a UX designer varies based on the situation but the goal remains the same: curate an excellent user experience.
Five Competencies of UX Design gives a very good overview of the broad areas that a UX designer works at. It is a must read to understand what can you expect if you are jumping into this field.
You can also have a look at What is a UX Designer which is another interpretation on what the role is all about.
Note: However, you can always look up different job descriptions on the company that is hiring. Many companies confuse the title of UX designer with the guy who develops the front-end of an application which is not what UX is all about. The job description should give you the best idea of whether you want to go for it or not.