Because of the increasing size and complexity of available graph structures in experimental sciences like molecular biology, techniques of graph visualization tend to reach their limit. To assist experimental scientists into the understanding of the underlying phenomena, most visualization methods are based on the organization of edges and nodes in clusters. Among recent ones, Power Graph Analysis is a lossless compression of the graph based on the search of cliques and bicliques, improving the readability of the overall structure. Royer et al. introduced a heuristic approach providing approximate solutions to this NP-complete problem. Later, Bourneuf et al. formalized the heuristic using Formal Concept Analysis. This paper proposes to extend this work by a for-malization of the graph compression search space. It shows that (1) the heuristic cannot always achieve an optimal compression, and (2) the concept lattice associated to a graph enables a more complete exploration of the search space. Our conclusion is that the search for graph compression can be usefully associated with the search for patterns in the concept lattice and that, conversely, confusing sets of objects and attributes brings new interesting problems for FCA.