The Elimination Procedure for the Competition Number is Not Optimal

Given an acyclic digraph D, the competition graph C(D) is defined to be the undirected graph with V (D) as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that the arcs (x, z) and (y, z) are both present in D. The competition number k(G) for an undirected graph G is the least number r such that there exists an acyclic digraph F on |V (G) | + r vertices where C(F) is G along with r isolated vertices. Kim and Roberts [3] introduced an elimination procedure for the competition number, and asked whether the procedure calculated the competition number for all graphs. We answer this question in the negative by demonstrating a graph where the elimination procedure does not calculate the competition number. This graph also provides a negative answer to a similar question about the related elimination procedure for the phylogeny number found in [2]. AMS Subject Classification: 05C20, 68R1