The concept of irredundance was introduced by Cockayne et al. (1978). Let denote the graph
neighborhood of a vertex in a graph (including itself), and let denote the union of neighborhoods for a set of vertices
. Then A set of vertices in a graph is called an irredundant set if, for every vertex ,

In other words, an irredundant set is a set of graph vertices such that the removal of any single vertex from the set gives a different union of neighborhoods than the union of neighborhood for the entire set.

An irredundant set of largest possible size is called a maximum irredundant set, and an irredundant set that is not a proper subset of a larger
irredundant set is called a maximal irredundant
set.

Any independent vertex set is an irredundant
set (Burger et al. 1997, Mynhardt and Roux 2020).

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Math.163, 47-66, 1997.Chartrand, G. and Lesniak, L. Graphs
& Digraphs, 4th ed. Boca Raton, FL: Chapman & Hall/CRC, pp. 286-287,
2005.Cockayne, E. J. Hedetniemi, S. T.; and Miller, D. J.
"Properties of Hereditary Hypergraphs and Middle Graphs." Canad. Math.
Bull.21< 461-468, 1978.Hedetniemi, S. T. and Laskar,
R. C. "A. Bibliography on Dominating Sets in Graphs and Some Basic Definitions
of Domination Parameters." Disc. Math.86, 257-277, 1990.Mynhardt,
C. M. and Roux, A. "Irredundance Graphs." 14 Apr. 2020. https://arxiv.org/abs/1812.03382.