We argue that the existing knowledge about modular decomposition of graphs and clan decomposition of 2-structures can be put to use advantageously in a context of data analysis. We show how to obtain visual descriptions of co-occurrence patterns by employing these decompositions on possibly generalized Gaifman graphs associated to datasets. We provide both theoretical advances that connect the proposed process to other data mining aspects (namely, closed set mining), as well as implemented algorithmics leading to an open-source tool that demonstrates our approach.