# On vertex adjacencies in the polytope of pyramidal tours with step-backs

## TL;DR

This paper characterizes vertex adjacencies in the polytope of pyramidal tours with step-backs, a special class of Hamiltonian cycles, providing a polynomial-time verifiable condition.

## Contribution

It introduces a necessary and sufficient condition for vertex adjacencies in the polytope's skeleton, advancing understanding of its combinatorial structure.

## Key findings

- Polynomial-time condition for vertex adjacency
- Characterization of the polytope's skeleton structure
- Enhanced understanding of pyramidal tours with step-backs

## Abstract

We consider the traveling salesperson problem in a directed graph. The pyramidal tours with step-backs are a special class of Hamiltonian cycles for which the traveling salesperson problem is solved by dynamic programming in polynomial time. The polytope of pyramidal tours with step-backs $PSB (n)$ is defined as the convex hull of the characteristic vectors of all possible pyramidal tours with step-backs in a complete directed graph. The skeleton of $PSB (n)$ is the graph whose vertex set is the vertex set of $PSB (n)$ and the edge set is the set of geometric edges or one-dimensional faces of $PSB (n)$. The main result of the paper is a necessary and sufficient condition for vertex adjacencies in the skeleton of the polytope $PSB (n)$ that can be verified in polynomial time.