with largest correlation under the topological constraint of fixed genus $G=0$. The genus is a topologically invariant property of a surface defined as the largest number of nonintersecting simple closed curves that can be drawn on the surface without separating it. Roughly speaking, it is the number of holes in a surface. The construction algorithm for such graph is: following the ordered list $S_{ord}$ starting from the couple of elements with largest correlation one adds a link between element $i$ and element $j$ if and only if the resulting graph can still be embedded on a plane or a sphere, i.e. topological surfaces with $G=0$. A basic difference of the PMFG with respect to the MST is the number of links which is $N-1$ in the MST and $3\,(N-2)$ in the PMFG. Moreover, the PMFG is a network with loops whereas the MST is a tree. It is worth recalling that Tumminello et al. (2005) have proven that the PMFG always contains the MST.

In Fig. \[pmfg10\] we show the PMFG obtained for the considered example. In the figure the length of the links is not related to the similarity measure between the two vertices they connect. We are using 