On $1$-skeleton of the polytope of pyramidal tours with step-backs

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 $\mathrm{PSB}(n)$ 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 $\mathrm{PSB}(n)$ 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 adjacency in the $1$-skeleton of the polytope $\mathrm{PSB}(n)$ and estimate the diameter and the clique number of the $1$-skeleton: the diameter is bounded above by $4$ and the clique number grows quadratically in the parameter $n$.