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This monogram considers certain sequential and parallel algorithms on interval graphs, permutation graphs and trapezoid graphs. These graphs arise quite naturally in real-world applications. I have tried to present a rigorous and coherent theory. Proofs are constructive and are streamlined as much as possible. I have directed much attention to the algorithmic aspects of every problem. Algorithms are expressed in a manner that will make their adaption to a particular programming language relatively easy. The complexity of every algorithm is analyzed so that some measure of its efficiency...