Decomposition by maxclique separators

We present a minimal counterexample to an O(|V||E|)O(|V||E|) algorithm for decomposing a graph G=(V,E)G=(V,E) by maximal cliques proposed by R. Tarjan. We also present a modification to this algorithm which is correct while retaining its O(|V||E|)O(|V||E|) complexity.