Longest Paths in Circular Arc Graphs

TL;DR

This paper addresses a gap in the proof of a theorem concerning the intersection of all longest paths in connected circular arc graphs, providing a complete and rigorous proof.

Contribution

It closes a gap in the existing proof of a key theorem about longest paths in circular arc graphs, ensuring its validity.

Findings

Confirmed the nonempty intersection of all longest paths in connected circular arc graphs.

Provided a complete proof resolving previous gaps in the theorem.

Strengthened the theoretical foundation of circular arc graph properties.

Abstract

As observed by Rautenbach and Sereni (arXiv:1302.5503) there is a gap in the proof of the theorem of Balister et al. (Longest paths in circular arc graphs, Combin. Probab. Comput., 13, No. 3, 311-317 (2004)), which states that the intersection of all longest paths in a connected circular arc graph is nonempty. In this paper we close this gap.