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A 3D Vertex Model has been successfully implemented to investigate the evolution of a special grain assembly during grain growth. The model considers the mean curvature as driving force for the motion of the vertices and allows the consideration of all parameters affecting the motion of the system, i.e., grain boundary energy and line tension of the triple lines, as well as grain boundary (GB), triple line (TL) and quadruple point (QP) mobility as well. The used special configuration makes it possible to study the influence of all structural elements of a grain boundary network on the evolution of the system by allowing the steady-state motion of the boundaries of a shrinking grain. In the present work the different mobilities have been systematically varied and the evolution of the grain size with time has been studied as a function of TL and QP mobility. The results of the simulations are finally linked to the different kinetic regimes reached by the system.