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A family of coordinates $\psi_h$ for the Teichm\"uller space of a compactsurface with boundary was introduced in \cite{l2}. In the work \cite{m1},Mondello showed that the coordinate $\psi_0$ can be used to produce a naturalcell decomposition of the Teichm\"uller space invariant under the action of themapping class group. In this paper, we show that the similar result also worksfor all other coordinate $\psi_h$ for any $h \geq 0$.

Cell decompositions of Teichm\"uller spaces of surfaces with boundary