Counting independent dominating sets in linear polymers

A vertex subset W subset of V of the graph G = (V, E) is an independent dominating set, if every vertex in V \ 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 (polymer graph) 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.