  The unordered configuration space of $n$ points on a graph $\Gamma,$ denoted
here by $UC^n(\Gamma),$ can be viewed as the space of all configurations of $n$
unlabeled robots on a system of one-dimensional tracks, which is interpreted as
a graph $\Gamma.$ The topology of these spaces is related to the number of
vertices of degree greater than 2; this number is denoted by $m(\Gamma).$ We
discuss a combinatorial approach to compute the topological complexity of a
"discretized" version of this space, $UD^n(\Gamma),$ and give results for
certain classes of graphs. In the first case, we show that for a large class of
graphs, as long as the number of robots is at least $2m(\Gamma)$, then
$TC(UD^n(\Gamma))=2m(\Gamma)+1.$ In the second, we show that as long as the
number of robots is at most half the number of vertex-disjoint cycles in
$\Gamma,$ we have $TC(UD^n(\Gamma))=2n+1.$

Algebraic Topology