I have recently started studying the cell decomposition of moduli spaces. Among the papers I read, I studied this paper, but there is something I do not understand and I can't find the answer on my own.
The cellular structure of $M_{0,4}$ is described at page 19, but I don't understand why the only Nakamura graphs corresponding to genus 0 and 4 poles are the three listed in Figure 8. For example, why is the following graph not taken into consideration? I think that the points in $M_{0,4}$ associated with this graph are all different from those associated with the three graphs in Figure 8. Indeed, in the former case the associated Riemann surfaces must have two poles with positive residuals, while in the latter case the Riemann surfaces have only one pole with positive residual.
More generally, why can we restrict to graphs with only one incoming pole and n-1 outgoing poles to describe the cellular structure of $M_{g,n}$? It seems to me that the points in $M_{g,n}$ which correspond to a Giddings-Wolpert differential with more than one pole with positive residual are not taken into consideration. So, what am I missing? I believe there is something I misunderstood, but I don't know what.
