  On a metric graph we introduce the notion of a free divisor as a replacement
for the notion of a base point free complete linear system on a curve. By means
of an example we show that the Clifford inequality is the only obstruction for
the existence of very special free divisors on a graph. This is different from
the situation of base point free linear systems on curves. It gives rise to the
existence of many types of divisors on graphs that cannot be lifted to curves
maintaining the rank and it also shows that classifications made for linear
systems of some fixed small positive Clifford index do not hold (exactly the
same) on graphs.
