Given an acyclic digraph D, the competition graph is defined to be the undirected graph with as its vertex set and where vertices x and y are adjacent if there exists another vertex z such that the arcs and are both present in D. The competition number for an undirected graph G is the least number r such that there exists an acyclic digraph F on vertices where 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].
Supported in part by a National Defense Science and Engineering Graduate Fellowship and National Science Foundation Grants EIA 0205116, DBI 9982983, and SBR 9709134.