On self-clique graphs with given clique sizes, II

The clique graph of a graph G is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques have a non-empty intersection. A graph is self-clique if it is isomorphic to its clique graph. We give a new characterization of the set of all connected self-clique graphs having all cliques but two of size 2.