Combinatorial Models and Polarity Control Rules in Heterocyclic Design Eugene V. Babaev Chemistry Department, Moscow State University, Moscow, 119899, Russia Abstract. The review is devoted to applications of combinatorial models for classification and design of heterocyclizations, recyclizations, and ring opening reactions of heterocycles. These models are based on the combination of concepts of reaction graphs and various polarity control rules. The existing models of reaction graphs and polar classifications of reagents are critically overviewed. The model of polar cyclization graphs is suggested for design of novel routes to the consonant heterocycles. The models of local and global polarity control in the heterocyclic ring opening reactions and their applications are discussed. The model of the ring bonds redistribution graphs for the recyclization reactions and its applications to the computer-assisted search of novel ring transformations are reviewed. Elaborated computer programs and examples of experimental confirmation of some predictions are discussed. 1. Introduction The ring with heteroatoms is the central subject of heterocyclic chemistry. Since the cycle is the fundamental topological concept, the processes of cycle formation and destruction are among the central topics in the chemistry of heterocycles. The reactions with such essential topological changes are the well- known processes of heterocyclizations (appearance of a cycle), ring opening reactions (disappearance of a cycle), and recyclizations (transformation of one cycle to another). Many generations of chemists have contributed to understanding peculiarities of these three fundamental types of reactions, but even nowadays the discovery of new examples of such reactions is occasional. Due to the lack of exact mathematical models the chemistry of heterocycles is rather an art than a strict science. Of course, there are many attempts to rationalize the logic of heterocyclic chemistry by applying various theoretical models and implement them as computer programs (see e.g., the comprehensive review 1 on this topic). In spite of these efforts, the spacious factual material on heterocyclic rings (trans)formation and cleavage still requires better classification. In our early works we suggested few combinatorial models helpful for classification and design of reactions in heterocyclic chemistry. These models, based on the combination of two important concepts of the reaction graphs and of the polarity control rules, have been applied to heterocyclizations, recyclizations and ring opening reactions of heterocycles. The goal of this paper is to review our methodology and to prove its usefulness in solving some selected problems of heterocyclic chemistry. The plan of the review is the following. In Section 2 we shall briefly review the concept of reaction graphs. Then in Section 3 we shall