# The Facets of the Subtours Elimination Polytope

## TL;DR

This paper provides a comprehensive description of the facets of the subtours elimination polytope for arbitrary graphs, extending previous results known for complete graphs, and offers a concise proof of this characterization.

## Contribution

It generalizes the facet characterization of the subtours elimination polytope to all graphs and presents a simplified proof of this result.

## Key findings

- Provides a minimal description of $P(G)$ for general graphs.
- Extends previous facet characterizations beyond complete graphs.
- Offers a concise proof of the new characterization.

## Abstract

Let $G=(V, E)$ be an undirected graph. The subtours elimination polytope $P(G)$ is the set of $x\in \mathbb{R}^E$ such that: $0\leq x(e)\leq 1$ for any edge $e\in E$, $x(\delta (v))=2$ for any vertex $v\in V$, and $x(\delta (U))\geq 2$ for any nonempty and proper subset $U$ of $V$. $P(G)$ is a relaxation of the Traveling Salesman Polytope, i.e., the convex hull of the Hamiton circuits of $G$. Maurras \cite{Maurras 1975} and Gr\"{o}tschel and Padberg \cite{Grotschel and Padberg 1979b} characterize the facets of $P(G)$ when $G$ is a complete graph. In this paper we generalize their result by giving a minimal description of $P(G)$ in the general case and by presenting a short proof of it.