A New Graph Parameter To Measure Linearity

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.