The competition number of a graph with exactly h holes, all of which are independent

AbstractGiven an acyclic digraph D, the competition graph C(D) of D is the graph with the same vertex set as D where two distinct vertices x and y are adjacent in C(D) if and only if there is a vertex v in D such that (x,v) and (y,v) are arcs of D. The competition number κ(G) of a graph G is the least number of isolated vertices that must be added to G to form a competition graph. The purpose of this paper is to prove that the competition number of a graph with exactly h holes, all of which are independent, is at most h+1. This generalizes the result for h=0 given by Roberts, and the result for h=1 given by Cho and Kim