Using extended resolution to represent strongly connected components of directed graphs

In this paper, we study how to represent a directed graph as a SAT problem. We study those directed graphs which consists of two strongly connected components (SCC). We reuse the SAT models which are known as the Black-and-White SAT representations. We present the so-called 3rd Solution Lemma: If a directed graph consists of two SCCs, A and B, and there is an edge from A to B, then the corresponding SAT representation has 3 solutions: the black assignment, the white assignment, and the 3rd solution can be written as not sign A union B. Using this result, we present an important negative result: We cannot represent all SAT problems as directed graphs using the Black-and-White SAT representations. Furthermore, we study the question how to represent an SCC by one Boolean variable to maintain the 3rd Solution Lemma. For that we use extended resolution.