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A graph $G$ is a cocomparability graph if there exists an acyclic transitiveorientation of the edges of its complement graph $\overline{G}$. LBFS$^{+}$ isa variant of the generic Lexicographic Breadth First Search (LBFS), which usesa 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 themaximum length of a cycle of vertex orderings of $G$ obtained via such asequence of LBFS$^{+}$ sweeps. Dusart and Habib conjectured in 2017 thatLexCycle($G$)=2 if $G$ is a cocomparability graph and proved it holds forinterval 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 disjointunion of $P_{2}$ and $P_{3}$. As corollaries, it's applicable for diamond-freecocomparability graphs, cocomparability graphs with girth at least 4, as wellas interval graphs.

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