Using maximality and minimality conditions to construct inequality chains

The following inequality chain has been extensively studied in the discrete mathematical literature: i r ~ y ~ i ~ f l ~ F ~IR, where ir and IR denote the lower and upper irredundance numbers of a graph, 2: and F denote the lower and upper domination numbers of a graph, i denotes the independent domination number and fl 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.