Discrete Mathematics
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The class of weakly triangulated comparability graphs and their complements are generalizations of interval graphs and chordal comparability graphs. We show that problems on these classes of graphs can be solved efficiently by transforming them into problems on chordal bipartite graphs. We show that recognition and independent set on weakly triangulated comparability graphs can be solved in O(n2) time in this manner, and that the number of weakly triangulated comparability graphs is $2^{\Theta ( n {{\log}^2} n)}$.\ We also give algorithms to compute transitive closure and transitive reduction in O(n2loglogn) time if the underlying undirected graph of the transitive closure is a weakly triangulated comparability graph.