A New Graph Parameter to Measure Linearity

Since its introduction to recognize chordal graphs by Rose, Tarjan, and Lueker, Lexicographic Breadth First Search (LexBFS) has been used to come up with simple, often linear time, algorithms on various classes of graphs. These algorithms, called multi-sweep algorithms, compute a number of LexBFS orderings σ1, . . . , σk, where σi is used to break ties for σi+1, we write LexBFS (σi) = σi+1. For instance, Corneil et al. gave a linear time multi-sweep algorithm to recognize interval graphs [SODA 1998], Kratsch et al. gave a certifying recognition algorithm for interval and permutation graphs [SODA 2003]. Since the number of LexBFS orderings for a graph is finite, after some fixed number of + sweeps, we will eventually loop in a sequence of σ1, . . . , σk vertex orderings such that σi+1 = LexBFS (σi) mod k. We study this new graph invariant, LexCycle(G), defined as the maximum length of a cycle of vertex orderings obtained via a sequence of LexBFS. In this work, we focus on graph classes with small LexCycle. We give evidence that a small LexCycle often leads to linear structure that has been exploited algorithmically on a number of graph classes. In particular, we show that for proper interval, interval, co-bipartite, domino-free cocomparability graphs, as well as trees, there exists two orderings σ and τ such that σ = LexBFS(τ ) and τ = LexBFS(σ). One of the consequences of these results is the simplest algorithm to compute a transitive orientation for these graph classes. It was conjectured by Stacho [2015] that LexCycle is at most the asteroidal number of the graph class, we disprove this conjecture by giving a construction for which LexCycle(G) > an(G), the asteroidal number of G.