CHAPTER 5 - Comparability Graphs

Abstract

This chapter discusses the class of perfect graphs known as comparability graphs or transitively orientable graphs; these graphs are discussed in connection with interval graphs. It also discusses the interaction between implication classes. In the process, the formula for the number t(G) of transitive orientations of a comparability graph G and a procedure is obtained for constructing them. The treatment follows Golumbic, in which most of the theory was developed. An alternate method for calculating t(G) appears in Shevrin and Filippov. The chapter describes an algorithm for calculating transitive orientations and for determining whether or not a graph is a comparability graph. This technique is a modification of one first presented by Pnueli, Lempel, and Even. A transitive orientation F could be constructed for a comparability graph G in 0(δe + n) steps, where δ is the maximum degree of a vertex, e is the number of edges, and n is the number of vertices. From the transitive orientation F, one can assign a minimum coloring of G using the height function in 0(n + e) additional steps. At the same time, a maximum clique could also be calculated.