  A complete characterization of the set of states that can be achieved through
Thermal Processes (TP) is given by describing all vertices, edges and facets of
the allowed set of states in the language of thermomajorization curves. TPs are
linked to transportation matrices, which leads to the existance of extremal TPs
that are not required in implemenation of any transition allowed by TPs, for
every dimension $d\geq 4$ of the state space. A property of the associated
graphs, biplanarity, which differentiates between these extremal TPs and the
necessary ones, is identified.
