A graph-theoretic discussion following up a recent paper in which Hamelink obtains an interesting sufficient condition for a graph to be a clique graph. In the present study related conditions are given that are necessary as well as sufficient. As an application of the result, it is shown that Hamelink's condition is also necessary in certain special cases and that here it can be greatly simplified. As another application, certain theorems are derived that are useful in practice in reducing the question of whether certain smaller or simpler graphs are clique graphs. Next, the clique graph results are related to some work of Fulkerson and Gross (RM-3984) on interval graphs and the consecutive 1's property. Finally, there are some remarks, motivated by the clique graph results, on graphs with no independent cut sets.
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