When applying graph searching algorithms, such as LexBFS, to order the vertices of a graph, one must occasionally break ties. Different tie-breaking rules have shed light on different structural properties of graphs. A rule in particular that's proven powerful is the $^+$ rule, which builds a sequence $\sigma_1, \sigma_2, \ldots$ of vertex orderings, where each ordering $\sigma_i$ is used to break ties for $\sigma_{i+1}$. As the total number of vertex orderings of a finite graph is finite, this sequence must end up in a cycle of vertex orderings. The possible length of this is the main subject of this work. Intuitively speaking, we prove for graphs with a known notion of linearity (e.g., interval graphs with their interval representation on the real line), this cycle cannot be too big, no matter which vertex ordering we start with. More precisely, Dusart and Habib conjectured that for a cocomparability graph the size of such a cycle will always be 2, independent of the starting order. Stacho then asked whether for an arbitrary graph, the size of such a cycle can be bounded by the asteroidal number of the graph. In this work while we answer the latter question negatively, we provide support for the former conjecture by proving it for the subclass of cocomparability graphs with no induced dominos. This subclass contains cographs, proper interval graphs, interval graphs and cobipartite graphs. We provide simpler independent proofs for each of these cases which lead to stronger results on these subclasses. Finally we prove that the same holds for trees.

翻译：当应用图形搜索算法, 如 LexBFS 等图形搜索算法, 以命令图形的顶端值时, 一个人必须偶尔断结 。 不同的断领规则会为图形的不同结构属性提供亮度。 特别被证明强大的一条规则是 $ $ $ 美元 的规则, 它创建了一个序列 $sigma_ 1, \ sigma_ 2, +ldots $ 顶端订单, 即每次订购 $\ sigma_ i$ 用于断断线 $\ sigma_ i $ combre 的连接。 由于限定图形的顶端命令的总数有限, 此序列必须结束于一个顶端点排序的循环的周期。 直径直线的直径直径, 直径直线的直径直线的直径直径直直直线图, 直径直直直径直径直径直直直直的直径直直直直直径直径直直的直径直径直直直直径直径直直直直直直直直直直直的直直直直直方, 直直的直直直直直直直直直直直直直直直直的直直直直直直直直直的直直直直直直直直直直直直直直直的直直直直的直直直直直直直直直直直直的直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直的直的直直的直的直的直直直直直的直直直直直的直直直直直直的直直的直直直直直直的直直直直的直直直直直直直直直直直直直直的直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直直