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Let \(D\) be an acyclic digraph. The competition graph of \(D\) has the 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-common enemy graph of \(D\) has the same set of vertices as \(D\) and an edge between vertices \(u\) and \(v\) if and only if there are vertices \(w\) and \(x\) in \(D\) such that \((w,u), (w,v), (u,x)\), and \((v,x)\) are arcs of \(D\). The competition number (respectively, double competition number) of a graph \(G\), denoted by \(k(G)\) (respectively, \(dk(G)\)), is the smallest number \(k\) such that \(G\) together with \(k\) isolated vertices is a competition graph (respectively, competition-common enemy graph) of an acyclic digraph.
It is known that \(dk(G) \leq k(G) + 1\) for any graph \(G\). In this paper, we give a sufficient condition under which a graph \(G\) satisfies \(dk(G) \leq k(G)\) and show that any connected triangle-free graph \(G\) with \(k(G) \geq 2\) satisfies that condition. We also give an upper bound for the double competition number of a connected triangle-free graph. Finally, we find an infinite family of graphs each member \(G\) of which satisfies \(k(G) = 2\) and \(dk(G) > k(G)\).