Towards A Unified View of Linear Structure on Graph Classes

Graph searching, a mechanism to traverse a graph in a spe- cific manner, is a powerful tool that has led to a number of elegant algo- rithms. This paper focuses on lexicographic breadth first search (LexBFS) (a variant of BFS) introduced in [8] to recognize chordal graphs. Since then, there has been a number of elegant LexBFS based, often multi-sweep, algorithms. These are algorithms that compute a sequence of LexBFS orderings {\sigma}1 , . . . , {\sigma}k , and use {\sigma}i to break ties when computing {\sigma}i+1 . The tie-breaking rule is often referred to as a + rule, and the graph search as LexBFS+ . We write LexBFS+ (G, {\sigma}i ) = {\sigma}i+1 . In this paper, we prove that LexBFS converges for a number of graph families, including interval graphs, cobipartite graphs, trees, and domino-free cocomparability graphs. In particular, we prove a fixed point theorem for LexBFS that shows the existence of two orderings {\sigma} and {\pi} where LexBFS+ (G, {\sigma}) = {\pi} and LexBFS+ (G, {\pi}) = {\sigma}. A consequence of this result, in particular, is the simplest algorithm to compute a transitive orientation for these graph classes, as well as providing a unified view of the linear structure seen on various graph families. In addition to this algorithmic consequence, we provide fixed point theorems for proper interval, interval, cobipartite, and domino-free cocomparability graphs, as well as trees. These are the first non-trivial results of this kind.