  In this paper we consider the generalized anchored configuration spaces on
$n$ labeled points on a~graph. These are the spaces of all configurations of
$n$ points on a~fixed graph $G$, subject to the condition that at least $q$
vertices in some pre-determined set $K$ of vertices of $G$ are included in each
configuration.
  We give a non-alternating formula for the Euler characteristic of such spaces
for arbitrary connected graphs, which are not trees. Furthermore, we completely
determine the homology groups of the generalized anchored configuration spaces
of $n$ points on a circle graph.
