The competition number of a graph whose holes do not overlap much

Let image be an acyclic digraph. The competition graph of image is a graph which has the same vertex set as image and has an edge between image and image if and only if there exists a vertex image in image such that image and image are arcs of image. For any graph image, image together with sufficiently many isolated vertices is the competition graph of some acyclic digraph. The competition number image of image is the smallest number of such isolated vertices.