  Pyramidal tours with step-backs are Hamiltonian tours of a special kind: the
salesperson starts in city 1, then visits some cities in ascending order,
reaches city $n$, and returns to city 1 visiting the remaining cities in
descending order. However, in the ascending and descending direction, the order
of neighboring cities can be inverted (a step-back). It is known that on
pyramidal tours with step-backs the traveling salesperson problem can be solved
by dynamic programming in polynomial time.
  We define the polytope of pyramidal tours with step-backs
$\operatorname{PSB}(n)$ as the convex hull of the characteristic vectors of all
possible pyramidal tours with step-backs in a complete directed graph. The
1-skeleton of $\operatorname{PSB}(n)$ is the graph whose vertex set is the
vertex set of the polytope, and the edge set is the set of geometric edges or
one-dimensional faces of the polytope. We present a linear-time algorithm to
verify vertex adjacencies in 1-skeleton of the polytope $\operatorname{PSB}(n)$
and estimate the diameter and the clique number of 1-skeleton: the diameter is
bounded above by 4 and the clique number grows quadratically in the parameter
$n$.
