Completeness is also straightforward, because, as discussed in Proposition 2, applying a LEFT-ARC transition to a configuration of H σ (C) or a RIGHT-ARC transition to a configuration of H β (C) will always generate a graph violating the SINGLE-HEAD constraint.
any graph that meets the SINGLE-HEAD constraint and can be obtained using the S P transition system (which has been proven complete) can also be generated by this one.