The images of the clique operator and its square are different (Q2922215)

Let \({\mathcal G}\) be the class of all graphs. The clique operator \(K: {\mathcal G} \rightarrow {\mathcal G}\) assigns to a graph \(G\) its clique graph, that is, the intersection graph of the cliques of \(G\). Let \(O_k\) be the graph obtained from \(K_{2k}\) by removing a perfect matching. It this paper it is proved that \(O_4\in K({\mathcal G})\) and that \(O_4\notin K^2({\mathcal G})\), answering in the negative the open problem whether \(K({\mathcal G}) = K^2({\mathcal G})\) holds. The first assertion is easy because \(K(O_3) = O_4\). The proof that \(O_4\notin K^2({\mathcal G})\) is more involved and reduces to proving that \(K^{-1}(O_4) \cap K({\mathcal G}) = \emptyset\).