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.
Categories: math.CO