Erratum to: "Remarks on the second neighborhood problem"

TL;DR

This paper identifies errors in previous proofs related to the second neighborhood problem in directed graphs and provides corrected proofs for the claims involving weighted tournaments and digraphs.

Contribution

It corrects and clarifies the proofs concerning weighted local median orders and digraphs derived from tournaments, ensuring mathematical accuracy.

Findings

Corrected proof of weighted local median order existence

Corrected proof for digraphs obtained from tournaments

Clarification of previous proof mistakes

Abstract

We prove that the proof of existence of weighted local median order of weighted tournaments is wrong and that the proof of the correct statement which asserts that every digraph obtained from a tournament by deleting a set of arcs incident to the same vertex contains a mistake, in the paper entitled "Remarks on the second neighborhood problem". We introduce correct proofs of each.