The complexity of comparability graph recognition and coloring

Using the notion of G-decomposition introduced in Golumbic [8, 9], we present an implementation of an algorithm which assigns a transitive orientation to a comparability graph in O(δ·|E|) time and O(|E|) space where δ is the maximum degree of a vertex and |E| is the number of edges. A quotient operation reducing the graph in question and preserving G-decomposition and transitive orientability is shown, and efficient solutions to a number of NP-complete problems which reduce to polynomial time for comparability graphs are discussed.