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A graph \(G\) is said to be maximal clique irreducible if each maximal clique in \(G\) contains an edge which is not contained in any other maximal clique of \(G\). In 1981, Opsut and Roberts proved that any interval graph is maximal clique irreducible. In this paper, we generalize their result and consider the question of characterizing maximal clique irreducible graphs.