Summary: For a finite or an infinite set \(X\), let \(2^X\) be the power set of \(X\). A class of simple graphs, called strong Boolean graphs, is defined on the vertex set \(2^X\setminus {X,\emptyset}\), with \(M\) adjacent to \(N\) if \(M \cap N=\emptyset\). In this paper, we characterize the annihilating-ideal graphs \(\mathbb{AG}(R) \) that are blow-ups of strong Boolean graphs, complemented graphs and pre-atomic graphs respectively. In particular, for a commutative ring \(R\) such that \(\mathbb{AG}(R)\) has a maximum clique \(S\) with \(3\leq |V(S)| \leq \infty\), we prove that \(\mathbb{AG}(R)\) is a blow-up of a strong Boolean graph if and only if it is a complemented graph, if and only if \(R\) is a reduced ring. If assume further that \(R\) is decomposable, then we prove that \(\mathbb{AG}(R)\) is a blow-up of a strong Boolean graph if and only if it is a blow-up of a pre-atomic graph. We also study the clique number and chromatic number of the graph \(\mathbb{AG}(R)\).