# The (l,m)-step Competition Number of a Graph

Presented at the *42nd International Conference on Combinatorics, Graph Theory and Computing*, Florida Atlantic University in Boca Raton, Florida, March 2011

#### Abstract

Let *dist _{D}(x,y)* be the number of arcs in the shortest directed path from

*x*to

*y*in digraph

*D*. The

*(l,m)*-step competition graph of

*D*, denoted by

*C*, is a graph with vertex set

_{(l,m)}(D)*V (D)*so that, for distinct vertices

*x*and

*y*, {

*x,y*} is an edge in

*C*if and only if for some vertex

_{(l,m)}(D)*z*in

*V (D)*,

*dist*≤

_{D-y}(x,z)*l*and

*dist*≤

_{D-x}(y,z)*m*. Here, we extend Fred Roberts' definition of the competition number

*k(G)*of a graph G, and introduce the

*(l,m)*-step competition number,

*k*. It is the minimum number

_{(l,m)}(G)*k*such that

*G*with

*k*isolated vertices is the

*(l,m)*-step competition graph of an acyclic digraph.

*This paper has been withdrawn.*