  The competition number k(G) of a graph G is the smallest number k such that G
together with k isolated vertices added is the competition graph of an acyclic
digraph. A chordless cycle of length at least 4 of a graph is called a hole of
the graph. The number of holes of a graph is closely related to its competition
number as the competition number of a graph which does not contain a hole is at
most one and the competition number of a complete bipartite graph $K_{\lfloor
\frac{n}{2} \rfloor, \lceil \frac{n}{2} \rceil}$ which has so many holes that
no more holes can be added is the largest among those of graphs with n
vertices. In this paper, we show that even if a connected graph G has many
holes, the competition number of G can be as small as 2 under some assumption.
In addition, we show that, for a connected graph G with exactly h holes and at
most one non-edge maximal clique, if all the holes of G are pairwise
edge-disjoint and the clique number $\omega = \omega (G)$ of G satisfies $2
\leq \omega \leq h+1$, then the competition number of G is at most $h - \omega
+ 3$.
