Irreducible (v3)(v3) configurations and graphs

Cubic bipartite graphs with girth at least 6 correspond to symmetric combinatorial (v3)(v3) configurations. In 1887, Martinetti described a simple construction method which enables one to construct all combinatorial (v3)(v3) configurations from a set of so-called irreducible configurations. The result has been cited several times since its publication, both in the sense of configurations and graphs. But after a careful examination, the list of irreducible configurations given by Martinetti has turned out to be incomplete. We will give the description of all irreducible configurations and corresponding graphs, including those which are missing in Martinetti's list.