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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 eachconfiguration.  We give a non-alternating formula for the Euler characteristic of such spacesfor arbitrary connected graphs, which are not trees. Furthermore, we completelydetermine the homology groups of the generalized anchored configuration spacesof $n$ points on a circle graph.

Homology and Euler characteristic of generalized anchored configuration spaces of graphs