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Let $D$ be an acyclic digraph. The competition graph of $D$ hasthe same set of vertices as $D$ and an edge between vertices $u$and $v$ if and only if there is a vertex $x$ in $D$ such that$(u,x)$ and $(v,x)$ are arcs of $D$. The competition number ofa graph $G$, denoted by $k(G)$, is the smallest number $k$ suchthat $G$ together with $k$ isolated vertices is the competitiongraph of an acyclic digraph. It is known to be difficult tocompute the competition number of a graph in general. Evencharacterizing the graphs with competition number one looks hard.In this paper, we continue the work done by Cho and Kim\cite{ck}to characterize the graphs with one hole and competition numberone. We give a sufficient condition for a graph with one hole tohave competition number one. This generates a huge class of graphswith one hole and competition number one. Then we completelycharacterize the graphs with one hole and competition number onethat do not have a vertex adjacent to all the vertices of thehole. Also we show that deleting pendant vertices from a connectedgraph does not change the competition number of the original graphas long as the resulting graph is not trivial, and this allows usto construct infinitely many graph having the same competitionnumber. Finally we pose an interesting open problem.
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