![]() Several people worked on different generalizations and variations of threshold graphs. We will not be surprised if new applications were discovered for these graphs, since they possess a rich and interesting structure. This book should be of interest to people from many different fields such as mathematical psychology, graph theory, parallel processing, resource allocations, scheduling theory, etc. We describe most of the applications of threshold graphs and their generalizations that are known. We bring together the works of all these people and present a unified approach to their methods. In both cases it turns out that the corresponding graphs are precisely the threshold graphs.Īll these examples show that threshold graphs have been used in various fields with different applications. Koren studied those degree sequences that are not convex combinations of other degree sequences others have studied graphs whose degree sequences “majorize” the degree sequences of all other graphs. Perhaps the earliest use of threshold graphs is in graph theory itself, namely in the study of degree sequences of graphs. Cogis has studied the so-called Ferrers digraphs, which are essentially the difference graphs. ![]() At the center of his proof was the class of graphs that he called chain graphs, which are nothing but difference graphs. Yannakakis proved in 1982 an important result in complexity theory, namely that recognizing whether or not a given poset has dimension at most three is NP-complete. Several results of these authors were discovered independently by others in graph theory. These biorders, it turns out, are simply the difference graphs. investigated a special kind of order relations called biorders, that were used in the study of learning behaviors of children. Koop studied cyclic scheduling and manpower allocation and found that the problem can be modeled using threshold graphs and threshold hypergraphs. Ecker and Zaks discovered these graphs independently and investigated them for their use in graph labeling as applied to open shop scheduling. Ordman found other uses for these graphs in resource allocation problems. More specifically they found that these graphs can be used to control the flow of information between processors, much like the traffic lights used in controlling the flow of traffic. Their discovery is motivated by applications in parallel processing. About the same time Henderson and Zalcstein discovered the same graphs and called them PV-chunk definable graphs. Most properties of threshold graphs can be easily translated to properties of difference graphs and vice versa.Ĭhvátal and Hammer coined the name “threshold graphs” and studied these graphs for their application in set- packing problems. Either these graphs, or a close kin of these graphs, the difference graphs, have been used implicitly in several other articles. ![]() In fact, these graphs were discovered independently and reported in different journals by people working in different areas. Threshold graphs play an important role in graph theory as well as in several applied areas such as psychology, computer science, scheduling theory, etc. In Annals of Discrete Mathematics, 1995 1.1 Motivation
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