Digital fundamental groups and edge groups of clique complexes

In previous work, we have defined—intrinsically, entirely within the digital setting—a fundamental group for digital images. Here, we show that this group is isomorphic to the edge group of the clique complex of the digital image considered as a graph. The clique complex is a simplicial complex and its edge group is well-known to be isomorphic to the ordinary (topological) fundamental group of its geometric realization. This identification of our intrinsic digital fundamental group with a topological fundamental group—extrinsic to the digital setting—means that many familiar facts about the ordinary fundamental group may be translated into their counterparts for the digital fundamental group: The digital fundamental group of any digital simple closed curve is Z; a version of the Seifert–van Kampen Theorem holds for our digital fundamental group; every finitely presented group occurs as the (digital) fundamental group of some digital image. We also show that the (digital) fundamental group of every 2D digital image is a free group.