The Preisach graph and longest increasing subsequences

The Preisach graph is a directed graph associated with a permutation $\rho\in{\cal S}_N$. We give an explicit bijection between its vertices and increasing subsequences of $\rho$ with the property that the length of a subsequence equals to the degree of nesting of the corresponding vertex inside a hierarchy of cycles and sub-cycles of the graph. As a consequence, the nesting degree of the Preisach graph equals the length of the longest increasing subsequence.