ABSTRACT
One of the major reasons why so much research has been done on domination in graphs is that the domination number appears in so many different mathematical contexts, or what we call frameworks. These include the contexts of: hypergraphs, or matrices of Os and ls; the algebraic solution to matrix equations of the form N · X ~ l; the generation of inequality chains of parameters; conditions defined on functions of the form f : V _, {0, 1}, or more generally functions of the form f : V ~ Y, for Y a subset of real numbers; conditions defined on subgraphs induced by subsets of vertices (S); conditions defined between a setS and its complement V - S; and even the mathematical study of the game of chess and related games. In this chapter we discuss each of these mathematical frameworks for the domination number of a graph, several of which are discussed in detail in other chapters and will only be reviewed here. In order to be self-contained, some definitions and results will be repeated.
