Let
be the number of dominating sets of size in a graph , then the domination polynomial of in the variable is defined as

where
is the (lower) domination number of (Kotek et al. 2012, Alikhani and Peng 2014).

is multiplicative over connected components (Alikhani and Peng 2014).

Precomputed dominations polynomials for many named graphs in terms of a variable
and in the Wolfram Language as GraphData[graph,
"DominationPolynomial"][x].

The following table summarizes closed forms for the domination polynomials of some common classes of graphs (cf. Alikhani and Peng 2014).

Alikhani, S. and Peng, Y.-H. "Dominating Sets and Domination Polynomial of Cycles." Global J. Pure Appl. Math.4, No. 2,
2008.Alikhani, S. and Peng, Y.-H. "Introduction to Domination Polynomial
of a Graph." Ars Combin.114, 257-266, 2014.Burger,
A. P.; Cockayne, E. J.; and Mynhardt, C. M. "Domination and Irredundance
in the Queens' Graph." Disc. Math.163, 47-66, 1997.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.Kotek, T.; Preen, J.; Simon, F.; Tittmann, P; and Trinks,
M. "Recurrence Relations and Splitting Formulas for the Domination Polynomial."
Electron. J. Combin.19, No. 3, Paper 47, 27 pp., 2012. http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i3p47.