Linear-time transitive orientation
Ross M. McConnell, Jeremy Spinrad · 1997
The transitive orientation problem is the problem of assigning a direction to each edge of a graph so that the resulting digraph is transitive. A graph is a comparability graph if such an assignment is possible. We describe an O(n + m) algorithm for the transitive orientation problem, where n and m are the number of vertices and edges of the graph; full details are given in [IS]. This gives linear time bounds for maximum clique and minimum vertex coloring on comparability graphs, recognition of two-dimensional partial orders, permutation graphs, cointerval graphs, and triangulated comparability graphs, and other combinatorial problems on comparability graphs and their complements.