An implicit representation of chordal comparability graphs in linear time

Abstract

Ma and Spinrad have shown that every transitive orientation of a chordal comparability graph is the intersection of four linear orders. That is, chordal comparability graphs are comparability graphs of posets of dimension four. Among other uses, this gives an implicit representation of a chordal comparability graph using O ( n ) integers so that, given two vertices, it can be determined in O ( 1 ) time whether they are adjacent, no matter how dense the graph is. We give a linear time algorithm for finding the four linear orders, improving on their bound of O ( n 2 ) .