Due to fast developments in data acquisition, we observe rapidly increasing amounts of data available in diverse areas. Simultaneously, we observe that in many applications the underlying data is inherently complex, making graphs a very useful and adequate data structure for formal representation. A large amount of graph based methods for pattern recognition have been proposed. Many of these methods actually rely on graph matching. In the present paper a novel encoding of graph matching information is proposed. The idea of this encoding is to formalize the stable cores of specific classes by means of graphs. In an empirical evaluation we show that it can be highly beneficial to focus on these stable parts of graphs during graph classification.