Upper Irredundance Number

The upper irredundance number  of a graph  is the maximum size of an irredundant set of vertices in . It is therefore equal to the size of a maximum irredundant set as well to the size of a maximal irredundant set since every maximum irredundant set is also maximal. The upper irredundance number is also equal to largest exponent in a irredundance polynomial.

The (lower) irredundance number may be similarly defined as the minimum size of a maximal irredundant set of vertices in  (Burger et al. 1997, Mynhardt and Roux 2020).

The lower irredundance number , lower domination number , lower independence number , upper independence number , upper domination number , and upper irredundance number  satsify the chain of inequalities

(Burger et al. 1997).

REFERENCES

Burger, A. P.; Cockayne, E. J.; and Mynhardt, C. M. "Domination and Irredundance in the Queens' Graph." Disc. Math. 163, 47-66, 1997.

Cockayne, E. J. and Mynhardt, C. M. "The Sequence of Upper and Lower Domination, Independence and Irredundance Numbers of a Graph." Disc. Math. 122, 89-102, 1993).

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.