We give a complete structural characterisation of the map the positive branch
of a one-way pattern implements. We start with the representation of the
positive branch in terms of the phase map decomposition, which is then further
analysed to obtain the primary structure of the matrix M, representing the
phase map decomposition in the computational basis. Using this approach we
obtain some preliminary results on the connection between the columns structure
of a given unitary and the angles of measurements in a pattern that implements
it. We believe this work is a step forward towards a full characterisation of
those unitaries with an efficient one-way model implementation.