  Let D be an acyclic digraph. The competition graph of D is a graph which has
the same vertex set as D and has an edge between 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. The competition number k(G) of G is the smallest
number of such isolated vertices.
  A hole of a graph is a cycle of length at least 4 as an induced subgraph. In
2005, Kim [5] conjectured that the competition number of a graph with h holes
is at most h+1. Though Li and Chang [8] and Kim et al. [7] showed that her
conjecture is true when the holes do not overlap much, it still remains open
for the case where the holes share edges in an arbitrary way. In order to share
an edge, a graph must have at least two holes and so it is natural to start
with a graph with exactly two holes. In this paper, the conjecture is proved
true for such a graph.
