Using maximality and minimality conditions to construct inequality chains

Abstract

The following inequality chain has been extensively studied in the discrete mathematical literature:

$$\text{if⩽y⩽i⩽β⩽Γ⩽}\text{IR}$$

, where ir and IR denote the lower and upper irredundance numbers of a graph, γ and Γ denote the lower and upper domination numbers of a graph, i denotes the independent domination number and β denotes the vertex independence number of a graph. More than one hundred papers have been published on aspects of this chain. In this paper we define a simple mechanism which explains why this inequality chain exists and how it is possible to define many similar chains of potentially arbitrary length.