  For a graph $G$, the first multiplicative Zagreb index $\prod_1$ is equal to
the product of squares of the vertex degrees, and the second multiplicative
Zagreb index $\prod_2$ is equal to the product of the products of degrees of
pairs of adjacent vertices. The (mutiplicative) Zagreb indices have been the
focus of considerable research in computational chemistry dating back to Gutman
and Trinajsti\'c in 1972.
  In this paper, we explore the mutiplicative Zagreb indices in terms of
arbitrary domination number. The sharp upper and lower bounds of $\prod_1(G)$
and $\prod_2(G)$ are given. In addition, the corresponding extreme graphs are
charaterized.

Combinatorics