On subgraphs of Cartesian product graphs and S-primeness

We consider S-prime graphs i.e., graphs which cannot be represented as a nontrivial subgraphs of a Cartesian product of two graphs. A characterization of S-prime graphs with connectivity 2 is given, providing a substantialy larger class of examples of such graphs as they were previously known. We then introduce a class of so-called simple S-prime graphs and a concept of extension of graphs, and prove a characterization of S-prime graphs via these concepts.