This article extends the uniqueness theory in (Park et al., 2021) and establishes general necessary and sufficient conditions for the uniqueness of $P$–$\Theta$ power flow solutions in an AC power system using some properties of the monotone regime and the power network topology. We show that the necessary and sufficient conditions can lead to tighter sufficient conditions for the uniqueness in several special cases. Our results are based on the existing notion of maximal girth and our new notion of maximal eye. Moreover, we develop a series–parallel reduction method and search-based algorithms for computing the maximal eye and the maximal girth, which are necessary for the uniqueness analysis. Reduction to a single line using the proposed reduction method is guaranteed for 2-vertex-connected series–parallel graphs. The relations between the parameters of the network before and after reduction are obtained. It is verified on real-world networks that the computation of the maximal eye can be reduced to the analysis of a much smaller power network, while the maximal girth is computed during the reduction process.