A compact formulation of the set of tours neither in a graph nor its complement is presented and illustrates a general methodology proposed for constructing polyhedral models of decision problems based upon permutations, projection and lifting techniques. Directed Hamilton tours on n vertex graphs are interpreted as (n-1)- permutations. Sets of extrema of Birkhoff polyhedra are mapped to tours neither in a graph nor its complement and these sets are embedded into disjoint orthogonal spaces as the solution set of a compact formulation. An orthogonal projection of its solution set into the subspace spanned by the Birkhoff polytope is the convex hull of all tours neither in a graph nor its complement. It’s suggested that these techniques might be adaptable for application to linear programming models of network and path problems.