Abstract

This chapter discusses the notion of perfection in weighted graphs—superperfection. In the process, a more general type of coloring the vertices of a graph are introduced, suggesting many interesting applications. The concept of superperfection is by Alan Hoffman and Ellis Johnson. They were motivated by the shipbuilding problem and most of the early results are theirs. There are two basic methods for demonstrating superperfection: providing a suitable coloring or giving a suitable acyclic orientation. It has been reported that superperfect graphs properly contain the comparability graphs. This leads one to ask under what conditions these two classes coincide. Therefore, the chapter provides one answer to this question and discusses some open problems. The chapter also relates the concept of superperfection to some ideas of linear programming.

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