 ${\Delta}(\Omega)$ (which is still under construction) such that $\deg v_0 = b(0)$ and $\deg^F v_0 = b(00)$.  For example, if $\tau_0({\Delta}) = |{\Delta}(0)_{\ge 0}|$, where $ |{\Delta}(0)_{\ge 0}|$ is the number of vertices in ${\Delta}$ of degree at least 0, and $b(0) \ge 0$, then $\tau_0({\Delta})$ is increased by 0. If $\tau_0({\Delta}) = |{\Delta}(0)^F_{[k_{0,0}, k_{0,0} ]}|$, where $|{\Delta}(0)^F_{[k_{0,0}, k_{0,0} ]}|$ is the number of vertices $