Simpler, Linear-Time Transitive Orientation via Lexicographic Breadth-First Search

Comparability graphs are the undirected graphs whose edges can be directed so that the resulting directed graph is transitive. They are related to posets and have applications in scheduling theory. This paper considers the problem of finding a transitive orientation of a comparability graph, a requirement for many of its applications. A linear-time algorithm is presented based on an elegant partition refinement scheme developed elsewhere for the problem. The algorithm is intended as a simpler and more practical alternative to the existing lineartime solution, which is commonly understood to be difficult and mainly of theoretical value. It accomplishes this by using Lexicographic Breadth-First Search to achieve the same effect as produced by modular decomposition in the earlier linear-time algorithm.