Irredundance

This chapter concerns irredundance in graphs, a concept introduced because it provides a certificate for a dominating set of a graph to be minimally dominating. Informally, a set X of vertices in a graph G is irredundant if each vertex in X dominates a vertex of G (perhaps itself) that is not dominated by any other vertex in X. We discuss the concepts of private neighbours, the private neighbour cube and generalised irredundance; mention the chain of lower and upper domination, independence and irredundance numbers; and discuss equality of parameters in the domination chain. We present bounds on irredundance parameters involving other graph parameters, including Nordhaus-Gaddum- and Gallai-type results. We review differences between and ratios of parameters in the domination chain, criticality and stability concepts, irredundance on chessboards and irredundant Ramsey numbers. Finally, we discuss reconfiguration of irredundant sets and complexity results.