# A construction for directed in-out subgraphs of optimal size

## TL;DR

This paper introduces a construction method for optimal size directed in-out subgraphs, providing bounds, optimality proofs, and applications to the traveling salesman problem, including planarity and edge minimization.

## Contribution

It presents a universal construction for k-in-out graphs that meets theoretical bounds and is optimal for even k, with applications to TSP problem conversions.

## Key findings

- Construction meets the lower bound for vertices in all cases.
- For even k, the construction is optimal in edges and planar.
- Provides constraints to aid TSP cutting-plane algorithms.

## Abstract

We discuss the recently introduced concept of k-in-out graphs, and provide a construction for k-in-out graphs for any positive integer k. We derive a lower bound for the number of vertices of a k-in-out graph for any positive integer k, and demonstrate that our construction meets this bound in all cases. For even k, we also prove our construction is optimal with respect to the number of edges, and results in a planar graph. Among the possible uses of in-out graphs, they can convert the generalized traveling salesman problem to the asymmetric traveling salesman problem, avoiding the "big M" issue present in most other conversions. We give constraints satisfied by all in-out graphs to assist cutting-plane algorithms in solving instances of traveling salesman problem which contain in-out graphs.