  In this paper we determine the topological complexity of configuration spaces
of graphs which are not necessarily trees, which is a crucial assumption in
previous results. We do this for two very different classes of graphs: fully
articulated graphs and banana graphs. We also complete the computation in the
case of trees to include configuration spaces with any number of points,
extending a proof of Farber. At the end we show that an unordered configuration
space on a graph does not always have the same topological complexity as the
corresponding ordered configuration space (not even when they are both
connected). Surprisingly, in our counterexamples the topological complexity of
the unordered configuration space is in fact smaller than for the ordered one.
