I do feel that a similar answer can be found here: https://workplace.stackexchange.com/a/2973/29909 but I can also see how it would not explain certain points about a person with a dual career path.
The short answer is that it is always good to tailor a resume and trim what is irrelevant to the position you are applying to. A few points that I think you will need to consider though
Pruning Jobs vs Employment Gaps
you are going to want to create a balance between removing or diminishing jobs that are irrelevant and making your resume look sparse. If you did 5 years in web, 10 in graphic, and 5 in web you will likely want to include something about that 10 year job in a resume as not including it would look strange.
Lateral vs Irrelevant Skills
Web development and graphics work are not necessarily mutually exclusive. Your graphics work may have helped you understand optimal resolutions for images on the web and how to adjust those images to save bandwidth in an application. You do need to trim the fat but be careful to leave in what could be considered a lateral skill in the field. Also make sure to present the lateral skill in the context of the job you are applying for.
Focused Career vs Growth Oriented
It is touched on in the answer I linked but just to rehash it here, you may want to show that you know more that just what the job calls for. Obviously having the right skills for a job is important but mentioning other skills shows you have interest in your own personal growth. Just make sure that it shows interest and does not eclipse relevant skills in your resume.