  The competition graph of a digraph D is a (simple undirected) 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. In general, it is hard to compute the
competition number k(G) for a graph G and it has been one of important research
problems in the study of competition graphs to characterize a graph by its
competition number. A hole of a graph is a cycle of length at least 4 as an
induced subgraph. It holds that the competition number of a graph cannot exceed
one plus the number of its holes if G satisfies a certain condition. In this
paper, we show that the competition number of a graph with exactly h holes any
two of which share at most one edge is at most h+1, which generalizes the
existing results on this subject.
