  A vertex subset $W\subseteq V$ of the graph $G = (V,E)$ is an independent
dominating set if every vertex in $V\setminus W$ is adjacent to at least one
vertex in $W$ and the vertices of $W$ are pairwise non-adjacent. We enumerate
independent dominating sets in several classes of graphs made by a linear or
cyclic concatenation of basic building blocks. Explicit recurrences are derived
for the number of independent dominating sets of these kind of graphs.
Generating functions for the number of independent dominating sets of
triangular and squares cacti chain are also computed.
