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The clique graph \(K(G)\) of a given graph \(G\) is the intersection graph of the collection of maximal cliques of \(G\). Given a family \(\mathcal{F}\) of graphs, the \emph{clique-inverse graphs} of \(\mathcal{F}\) are the graphs whose clique graphs belong to \(\mathcal{F}\). In this work, we describe characterizations for clique-inverse graphs of bipartite graphs, chordal bipartite graphs, and trees. The characterizations lead to polynomial time algorithms for the corresponding recognition problems.