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The article introduces a representation of a formal context by an undirected graph called a context graph with the formal objects being the nodes of the graph. We use as a defining property for this graph that it contains every concept extent as a connected subgraph. The graph is not uniquely defined by this property - we focus on those graphs that are edge-minimal and present a result with respect to the number of their edges. We then study how the structure of an edge-minimal context graph can be updated to adjust to the subsequent addition of an object to the context. This leads to an incremental construction algorithm that does not require the explicit computation of formal concepts.