A Note on Improved bounds for the Oriented Radius of Mixed Multigraphs

TL;DR

This paper revises algorithms and bounds related to the oriented radius of mixed multigraphs after identifying and correcting an error in previous foundational observations.

Contribution

It corrects an error in earlier algorithms and bounds for the oriented radius of mixed multigraphs, ensuring the validity of the results.

Findings

Corrected algorithms for orienting mixed multigraphs.

Validated bounds for the oriented radius after correction.

Ensured the correctness of previous theoretical bounds.

Abstract

For a positive integer $r$, let $f(r)$ denote the smallest number such that any 2-edge connected mixed graph with radius $r$ has an oriented radius of at most $f(r)$. Recently, Babu, Benson, and Rajendraprasad significantly improved the upper bound of $f(r)$ by establishing that $f(r) \leq 1.5r^2 + r + 1$, see [Improved bounds for the oriented radius of mixed multigraphs, J. Graph Theory, 103 (2023), 674-689]. Additionally, they demonstrated that if each edge of a graph $G$ is contained within a cycle of length at most $\eta$, then the oriented radius of $G$ is at most $1.5r\eta$. The authors' results were derived through Observation 1, which served as the foundation for the development of Algorithm ORIENTOUT and Algorithm ORIENTIN. By integrating these algorithms, they obtained the improved bounds. However, an error has been identified in Observation 1, necessitating revisions to Algorithm ORIENTOUT and Algorithm ORIENTIN. In this note, we address the error and propose the necessary modifications to both algorithms, thereby ensuring the correctness of the conclusions.