  This paper investigates a generalized hyperbolic circle packing (including
circles, horocycles or hypercycles) with respect to the total geodesic
curvatures on the surface with boundary. We mainly focus on the existence and
rigidity of circle packing whose contact graph is the $1$-skeleton of a finite
polygonal cellular decomposition, which is analogous to the construction of
Bobenko and Springborn [4]. Motivated by Colin de Verdi\`ere's method [6], we
introduce the variational principle for generalized hyperbolic circle packings
on polygons. By analyzing limit behaviours of generalized circle packings on
polygons, we give an existence and rigidity for the generalized hyperbolic
circle packing with conical singularities regarding the total geodesic
curvature on each vertex of the contact graph. As a consequence, we introduce
the combinatoral Ricci flow to find a desired circle packing with a prescribed
total geodesic curvature on each vertex of the contact graph.
