The competition number of a graph with exactly [formula omitted] holes, all of which are independent

Abstract

Given 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.