  In this paper, we study the existence and rigidity of (degenerated) circle
pattern metric with prescribed total geodesic curvatures in spherical
background geometry. To find the (degenerated) circle pattern metric with
prescribed total geodesic curvatures, we define some prescribed combinatorial
Ricci flows and study the convergence of flows for (degenerated) circle pattern
metrics. We solve the prescribed total geodesic curvature problem and provide
two methods to find the degenerated circle pattern metric with prescribed total
geodesic curvatures. As far as we know, this is the first degenerated result
for total geodesic curvatures in spherical background geometry.
