  We focus on the algorithm underlying the main result of [A. Mestre, R. Oeckl,
Generating loop graphs via Hopf algebra in quantum field theory. J. Math.
Phys., 47, 122302, 2006]. This is an algebraic formula to generate all
connected graphs in a recursive and efficient manner. The key feature is that
each graph carries a scalar factor given by the inverse of the order of its
group of automorphisms.
  In the present paper, we revise that algorithm on the level of graphs.
Moreover, we extend the result subsequently to further classes of connected
graphs, namely, (edge) biconnected, simple and loopless graphs. Our method
consists of basic graph transformations only.
