# The LexCycle on $\overline{P_{2}\cup P_{3}}$-free Cocomparability Graphs

## TL;DR

This paper proves that for a specific class of cocomparability graphs, the LexCycle length is always 2, confirming a conjecture for these graph classes and extending previous results.

## Contribution

It establishes that LexCycle(G)=2 for -free cocomparability graphs, including diamond-free and girth  graphs, generalizing prior work on interval graphs.

## Key findings

- LexCycle(G)=2 for -free cocomparability graphs
- Applicable to diamond-free and girth  cocomparability graphs
- Confirms Dusart and Habib's conjecture for these classes

## Abstract

A graph $G$ is a cocomparability graph if there exists an acyclic transitive orientation of the edges of its complement graph $\overline{G}$. LBFS$^{+}$ is a variant of the generic Lexicographic Breadth First Search (LBFS), which uses a specific tie-breaking mechanism. Starting with some ordering $\sigma_{0}$ of $G$, let $\{\sigma_{i}\}_{i\geq 1}$ be the sequence of orderings such that $\sigma_{i}=$LBFS$^{+}(G, \sigma_{i-1})$. The LexCycle($G$) is defined as the maximum length of a cycle of vertex orderings of $G$ obtained via such a sequence of LBFS$^{+}$ sweeps. Dusart and Habib conjectured in 2017 that LexCycle($G$)=2 if $G$ is a cocomparability graph and proved it holds for interval graphs. In this paper, we show that LexCycle($G$)=2 if $G$ is a $\overline{P_{2}\cup P_{3}}$-free cocomparability graph, where a $\overline{P_{2}\cup P_{3}}$ is the graph whose complement is the disjoint union of $P_{2}$ and $P_{3}$. As corollaries, it's applicable for diamond-free cocomparability graphs, cocomparability graphs with girth at least 4, as well as interval graphs.