On second iterated clique graphs that are also third iterated clique graphs

Iterated clique graphs arise when the clique operator is applied to a graph more than once. Determining whether a graph is a clique graph or an iterated clique graph is usually a difficult task. The fact that being a clique graph and being an iterated clique graph are not equivalent things has been proved recently. However, it is still unknown whether the classes of second iterated clique graphs and third iterated clique graphs are the same. In this work we find classes of graphs, defined by means of conditions on the clique size and the structure of the clique intersections, whose second iterated clique graphs are also third iterated clique graphs.