  The notion of a competition graph was introduced by J. E. Cohen in 1968. The
competition graph C(D) of a digraph $D$ is a (simple undirected) graph which
has the same vertex set as D and has an edge between two distinct vertices x
and y if and only if there exists a vertex v in D such that (x,v) and (y,v) are
arcs of D. For any graph G, G together with sufficiently many isolated vertices
is the competition graph of some acyclic digraph. In 1978, F. S. Roberts
defined the competition number k(G) of a graph G as the minimum number of such
isolated vertices. In general, it is hard to compute the competition number
k(G) for a graph G and it has been one of the important research problems in
the study of competition graphs to characterize a graph by its competition
number. In 1982, R. J. Opsut gave two lower bounds for the competition number
of a graph. In this paper, we give a generalization of these two lower bounds
for the competition number of a graph.
