CHAPTER 10 - Threshold Graphs

Abstract

This chapter discusses a particularly simple technique for distinguishing between stable and nonstable subsets of vertices in a special class of graphs. The graphs that admit this technique, which involves assigning certain weights to the vertices, are called threshold graphs. Threshold graphs were introduced by Chvátal and Hammer. Threshold graph is a split graph as its vertices can be partitioned into a stable set and a complete set. Secondly, the edges of threshold can be transitively oriented. The complement of threshhold graph can also be transitively oriented;therefore, threshold graph is a special kind of permutation graph. Threshold graphs were rediscovered and studied by others, including Henderson and Zalcstein; they are responsible for the application of threshold graph for synchronizing parallel processes.