A Cleanup on Transitive Orientation

In the past, diierent authors developed distinct approaches to the problem of transitive orientation. This also resulted in diierent ideas and diierent theorems which seem unrelated. In this paper we show the connections between these theories and present a new algorithm to recognize a comparability graph. A comparability graph is an undirected graph G = (V; E), jV j = n, jEj = m, in which every edge may be assigned a direction so that the resulting digraph is a partial order. To the best of our knowledge, literature so far knows mainly two solutions for this problem. The rst solution is due to Golumbic 7] and computes a G-decomposition recursively. This is a partition of the edges into so-called implication classes which de-ne a transitive orientation. This algorithm runs in time O(n m). The resulting orientation is transitive if the algorithm terminates successfully. The second solution with running time O(n 2) was given by Spinrad 11]. This algorithm computes a transitive orientation if there is such an orientation at all. This is a fundamental diierence to Golumbic's algorithm. Whereas Golumbic's algorithm nds a transitive orientation when it terminates successfully, Spinrad's algorithm always computes an orientation of the graph. This orientation is transitive if there is such an orientation at all, however this has to be tested separately. The diicult part of Spinrad's method is the computation of the input data structure for the proper orientation process, the so-called Modular Decomposition, a recursively deened tree-representation of G. A restricted class of the Modular Decomposition, denoted as cotrees, arises in the context of cographs. In 2] an incremental algorithm is presented to construct a cotree in linear time. Our algorithm is a natural extension of the cograph recognition algorithm in 2], it computes simultaneously a transitive orientation and a Modular Decomposition of a graph. The extension makes use of the condition which makes the cograph recognition algorithm fail. The main result is a structural improvement of the theory of comparability graphs. Our algorithm takes O(n 2) time, which is a lower bound for algorithms relying upon a matrix representation of a graph. We show that modular decomposition and G-decomposition are not independent structures, they imply each other. Based on this perception, we develop new, shorter proofs of the algorithmically relevant theorems about comparability graphs and easily veriiable proofs for the correctness of the algorithms of Spinrad and Golumbic. Especially we show …