On the skeleton of the pyramidal tours polytope

TL;DR

This paper analyzes the skeleton of the pyramidal tours polytope, providing conditions for vertex adjacency, an efficient algorithm, and key properties like diameter and clique number, advancing understanding of its geometric and combinatorial structure.

Contribution

It offers a complete characterization of vertex adjacency in the pyramidal tours polytope and introduces a linear-time algorithm for adjacency checking, along with fundamental properties of the polytope's skeleton.

Findings

Vertex adjacency characterized by necessary and sufficient conditions.

Developed a linear complexity algorithm for adjacency checking.

Proved the diameter of the skeleton is 2 and estimated the clique number as Θ(n^2).

Abstract

We consider the skeleton of the pyramidal tours polytope. Hamiltonian tour is called pyramidal if the salesperson starts in city $1$, then visits some cities in increasing order, reaches city $n$ and returns to city $1$, visiting the remaining cities in decreasing order. The polytope $PYR(n)$ is defined as the convex hull of characteristic vectors of all pyramidal tours in the complete graph $K_{n}$. The skeleton of the polytope $PYR(n)$ is the graph whose vertex set is the vertex set of $PYR(n)$ and edge set is the set of geometric edges or one-dimensional faces of $PYR(n)$. We describe the necessary and sufficient condition for the adjacency of vertices of the polytope $PYR(n)$. On this basis we developed an algorithm to check the vertex adjacency with a linear complexity. We establish that the diameter of $PYR(n)$ skeleton equals 2, and the asymptotically exact estimate of $PYR(n)$ skeleton's clique number is $\Theta(n^{2})$. It is known that this value characterizes the time complexity in a broad class of algorithms based on linear comparisons.