The competition number of a graph having exactly one hole Original Research Article

Let D be an acyclic digraph. The competition graph of D has the same set of vertices as D and an edge between vertices u and image if and only if there is a vertex x in D such that image and image are arcs of D. The competition number of a graph G, denoted by image, is the smallest number k such that G together with k isolated vertices is the competition graph of an acyclic digraph. In this paper, we show that the competition number of a graph having exactly one chordless cycle of length at least 4 is at most two. We also give a large family of such graphs whose competition numbers are less than or equal to one.