A minimum dominating set is a dominating set of smallest size in a given graph. The size of a minimum dominating set is known as the domination number of the graph.
A minimum dominating set is always a minimal dominating set, but the converse does not necessarily hold.
Finding a minimum dominating set of a general graph is NP-complete, which can be shown by reduction from the vertex cover problem
(Garey and Johnson 1983, Mertens 2024). This means that no polynomial-time algorithm
exists to compute a minimum dominating set. The fastest known algorithm to find a
minimum dominating set for a general graph with vertex
count 
has time complexity 
(van Rooij and Bodlaender 2011, Mertens 2024).
