Independence Related Graph Theory Parameters The chain of related parameters ir(G) ≤ γ(G) ≤ i(G) ≤ β(G) ≤ Γ(G) ≤ IR(G) has been widely studied by graph theorists. The work in this dissertation follows from a series of attempts to find a set of reasonable parameters that lie between the independence parameters i(G) and β(G) in this chain. During and after this effort, several other results of interest were found relating to the concept of "knockouts" in a graph. A related operation of "replacement" is produced and it is shown that these concepts allow the construction of maximal independent subsets of a graph. These concepts allow graph properties with an associated parameter pair between i(G) and β(G) to produce a good heuristic for the calculation ofβ(G) for many graphs, and to produce a number of other properties which seem to be of interest.