Switchings, extensions, and reductions in central digraphs

A directed graph is called central if its adjacency matrix A satisfies the equation A 2 = J , where J is the matrix with a 1 in each entry. It has been conjectured that every central directed graph can be obtained from a standard example by a sequence of simple operations called switchings, and also that it can be obtained from a smaller one by an extension. We disprove these conjectures and present a general extension result which, in particular, shows that each counterexample extends to an infinite family.