  Domination is the fastest-growing field within graph theory with a profound
diversity and impact in real-world applications, such as the recent
breakthrough approach that identifies optimized subsets of proteins enriched
with cancer-related genes. Despite its conceptual simplicity, domination is a
classical NP-complete decision problem which makes analytical solutions elusive
and poses difficulties to design optimization algorithms for finding a
dominating set of minimum cardinality in a large network. Here we derive for
the first time an approximate analytical solution for the density of the
minimum dominating set (MDS) by using a combination of cavity method and
Ultra-Discretization (UD) procedure. The derived equation allows us to compute
the size of MDS by only using as an input the information of the degree
distribution of a given network.
