The elimination procedure for the competition number is not optimal

Given an acyclic digraph D, the competition graph image is defined to be the undirected graph with image as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that the arcs image and image are both present in D. The competition number image for an undirected graph G is the least number r such that there exists an acyclic digraph F on image vertices where image is G along with r isolated vertices. Kim and Roberts [The Elimination Procedure for the Competition Number, Ars Combin. 50 (1998) 97–113] 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 introduced by the current author in [S.G. Hartke, The Elimination Procedure for the Phylogeny Number, Ars Combin. 75 (2005) 297–311].