The hypermetric cone and polytope on graphs

The hypermetric cone was defined in [9] and was extensively studied by Michel Deza and his collaborators. Another key interest of him was cut and metric polytope which he considered in his last works in the case of graphs.
Here we combine both interest by considering the hypermetric on graphs. We define them for any graph and give an algorithm for computing the extreme rays and facets of hypermetric cone on graphs. We compute the hypermetric cone for the first non-trivial case of $K_7 - \{e\}$. We also compute the hypermetric cone in the case of graphs with no $K_5$ minor.