Are you cutting the graphics out of a single color of vinyl or other material? If so, you'll need to look at the artwork in Outline View and use Pathfinder operations such as Unite to remove overlapping paths. The plotter blade is going to cut wherever a path is present. Even some variable fonts have overlapping paths that must be welded before being cut in a vinyl plotter or on a routing table.
Applications that do both printing and cutting often require the paths to have a custom-named spot color applied to the line strokes. The spot color name is most often "CutContour" but it can vary depending on the application. A regular cutting-only application shouldn't require those steps. You may have to check the documentation of the cutting application you're using.